CINXE.COM

סימון אסימפטוטי – ויקיפדיה

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-disabled skin-theme-clientpref-day vector-toc-available" lang="he" dir="rtl"> <head> <meta charset="UTF-8"> <title>סימון אסימפטוטי – ויקיפדיה</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-disabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )hewikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy" ,"wgMonthNames":["","ינואר","פברואר","מרץ","אפריל","מאי","יוני","יולי","אוגוסט","ספטמבר","אוקטובר","נובמבר","דצמבר"],"wgRequestId":"0948f6ed-b621-4830-90e6-7781fccff81b","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"סימון_אסימפטוטי","wgTitle":"סימון אסימפטוטי","wgCurRevisionId":39136980,"wgRevisionId":39136980,"wgArticleId":274565,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["סימנים מתמטיים","אלגוריתמים","סיבוכיות חישובית"],"wgPageViewLanguage":"he","wgPageContentLanguage":"he","wgPageContentModel":"wikitext","wgRelevantPageName":"סימון_אסימפטוטי","wgRelevantArticleId":274565,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia", "wgCiteReferencePreviewsActive":true,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"he","pageLanguageDir":"rtl","pageVariantFallbacks":"he"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":10000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q269878","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false,"wgSiteNoticeId":"2.254"};RLSTATE={ "ext.gadget.ExternalLinkIcon":"ready","ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","mediawiki.codex.messagebox.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.relatedArticles.styles":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready","ext.dismissableSiteNotice.styles":"ready"};RLPAGEMODULES=["mediawiki.page.media","site","mediawiki.page.ready","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.EditToolbar","ext.gadget.TemplateParamWizard","ext.gadget.Summarieslist","ext.gadget.refsToLeft","ext.gadget.Wdsearch","ext.gadget.catSexSelector","ext.gadget.feedback","ext.gadget.wikibugs", "ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.relatedArticles.readMore.bootstrap","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking","ext.dismissableSiteNotice"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=he&amp;modules=ext.dismissableSiteNotice.styles%7Cext.math.styles%7Cext.relatedArticles.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cmediawiki.codex.messagebox.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&amp;only=styles&amp;printable=1&amp;skin=vector-2022"> <script async="" src="/w/load.php?lang=he&amp;modules=startup&amp;only=scripts&amp;printable=1&amp;raw=1&amp;skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=he&amp;modules=ext.gadget.ExternalLinkIcon&amp;only=styles&amp;printable=1&amp;skin=vector-2022"> <link rel="stylesheet" href="/w/load.php?lang=he&amp;modules=site.styles&amp;only=styles&amp;printable=1&amp;skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="noindex,follow,max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="סימון אסימפטוטי – ויקיפדיה"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//he.m.wikipedia.org/wiki/%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99"> <link rel="alternate" type="application/x-wiki" title="עריכה" href="/w/index.php?title=%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99&amp;action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="ויקיפדיה"> <link rel="EditURI" type="application/rsd+xml" href="//he.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://he.wikipedia.org/wiki/%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.he"> <link rel="alternate" type="application/atom+xml" title="הזנת Atom של ויקיפדיה" href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%A9%D7%99%D7%A0%D7%95%D7%99%D7%99%D7%9D_%D7%90%D7%97%D7%A8%D7%95%D7%A0%D7%99%D7%9D&amp;feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki rtl sitedir-rtl mw-hide-empty-elt ns-0 ns-subject mw-editable page-סימון_אסימפטוטי rootpage-סימון_אסימפטוטי skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">לדלג לתוכן</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="אתר"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="תפריט ראשי" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">תפריט ראשי</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">תפריט ראשי</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">העברה לסרגל הצד</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">הסתרה</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> ניווט </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage" class="mw-list-item"><a href="/wiki/%D7%A2%D7%9E%D7%95%D7%93_%D7%A8%D7%90%D7%A9%D7%99" title="ביקור בעמוד הראשי [z]" accesskey="z"><span>עמוד ראשי</span></a></li><li id="n-welcomepage" class="mw-list-item"><a href="/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:%D7%91%D7%A8%D7%95%D7%9B%D7%99%D7%9D_%D7%94%D7%91%D7%90%D7%99%D7%9D"><span>ברוכים הבאים</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%A9%D7%99%D7%A0%D7%95%D7%99%D7%99%D7%9D_%D7%90%D7%97%D7%A8%D7%95%D7%A0%D7%99%D7%9D" title="רשימת השינויים האחרונים באתר [r]" accesskey="r"><span>שינויים אחרונים</span></a></li><li id="n-featured-articles" class="mw-list-item"><a href="/wiki/%D7%A4%D7%95%D7%A8%D7%98%D7%9C:%D7%A2%D7%A8%D7%9B%D7%99%D7%9D_%D7%9E%D7%95%D7%9E%D7%9C%D7%A6%D7%99%D7%9D"><span>ערכים מומלצים</span></a></li><li id="n-portals" class="mw-list-item"><a href="/wiki/%D7%A4%D7%95%D7%A8%D7%98%D7%9C:%D7%A4%D7%95%D7%A8%D7%98%D7%9C%D7%99%D7%9D"><span>פורטלים</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%90%D7%A7%D7%A8%D7%90%D7%99" title="טעינת דף אקראי [x]" accesskey="x"><span>ערך אקראי</span></a></li> </ul> </div> </div> <div id="p-community" class="vector-menu mw-portlet mw-portlet-community" > <div class="vector-menu-heading"> קהילה </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-portal" class="mw-list-item"><a href="/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:%D7%A9%D7%A2%D7%A8_%D7%94%D7%A7%D7%94%D7%99%D7%9C%D7%94" title="אודות המיזם, איך אפשר לעזור, איפה למצוא דברים"><span>שער הקהילה</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/%D7%A2%D7%96%D7%A8%D7%94:%D7%AA%D7%A4%D7%A8%D7%99%D7%98_%D7%A8%D7%90%D7%A9%D7%99" title="המקום למצוא מידע"><span>עזרה</span></a></li><li id="n-Ask-q" class="mw-list-item"><a href="/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:%D7%99%D7%99%D7%A2%D7%95%D7%A5" title="דפים להתייעצות טכנית, לשונית וכו&#039;"><span>ייעוץ</span></a></li><li id="n-village-pump" class="mw-list-item"><a href="/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:%D7%9E%D7%96%D7%A0%D7%95%D7%9F" title="דיוני מדיניות ודיונים כלליים על ויקיפדיה"><span>מזנון</span></a></li><li id="n-Town-square" class="mw-list-item"><a href="/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:%D7%9B%D7%99%D7%9B%D7%A8_%D7%94%D7%A2%D7%99%D7%A8" title="המקום בוויקיפדיה לשתף את הקהילה בדעותיכם בנושאים שאינם קשורים ישירות לוויקיפדיה"><span>כיכר העיר</span></a></li><li id="n-news" class="mw-list-item"><a href="/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:%D7%97%D7%93%D7%A9%D7%95%D7%AA" title="חדשות בנושאי ויקיפדיה שונים"><span>חדשות</span></a></li><li id="n-notice-board" class="mw-list-item"><a href="/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:%D7%9C%D7%95%D7%97_%D7%9E%D7%95%D7%93%D7%A2%D7%95%D7%AA" title="לוח מודעות להפניית הוויקיפדים לדיונים בדפי שיחה שלא זכו לתשומת לב מספקת"><span>לוח מודעות</span></a></li><li id="n-contactUs" class="mw-list-item"><a href="/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:%D7%99%D7%A6%D7%99%D7%A8%D7%AA_%D7%A7%D7%A9%D7%A8" title="איך ליצור קשר עם ויקיפדיה לצורך תיקונים, בירורים או שאלות אחרות"><span>יצירת קשר</span></a></li><li id="n-guestbook" class="mw-list-item"><a href="/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:%D7%A1%D7%A4%D7%A8_%D7%90%D7%95%D7%A8%D7%97%D7%99%D7%9D" title="שתפו אותנו במשוב כללי על התרשמותכם מוויקיפדיה ומידת שביעות רצונכם ממנה"><span>ספר אורחים</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/%D7%A2%D7%9E%D7%95%D7%93_%D7%A8%D7%90%D7%A9%D7%99" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="ויקיפדיה" src="/static/images/mobile/copyright/wikipedia-wordmark-he.svg" style="width: 7.5em; height: 1.75em;"> <img class="mw-logo-tagline" alt="האנציקלופדיה החופשית" src="/static/images/mobile/copyright/wikipedia-tagline-he.svg" width="120" height="12" style="width: 7.5em; height: 0.75em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%97%D7%99%D7%A4%D7%95%D7%A9" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="‏חיפוש בוויקיפדיה‏ [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>חיפוש</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="חיפוש בוויקיפדיה" aria-label="חיפוש בוויקיפדיה" autocapitalize="sentences" title="‏חיפוש בוויקיפדיה‏ [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="מיוחד:חיפוש"> </div> <button class="cdx-button cdx-search-input__end-button">חיפוש</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="כלים אישיים"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="מראה"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="שינוי המראה של גודל הגופן, הרוחב והצבע של הדף" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="מראה" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">מראה</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&amp;utm_medium=sidebar&amp;utm_campaign=C13_he.wikipedia.org&amp;uselang=he" class=""><span>תרומה לוויקיפדיה</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A8%D7%A9%D7%9E%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&amp;returnto=%D7%A1%D7%99%D7%9E%D7%95%D7%9F+%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99&amp;returntoquery=printable%3Dyes" title="מומלץ ליצור חשבון ולהיכנס אליו, אך אין חובה לעשות זאת" class=""><span>יצירת חשבון</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%9B%D7%A0%D7%99%D7%A1%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&amp;returnto=%D7%A1%D7%99%D7%9E%D7%95%D7%9F+%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99&amp;returntoquery=printable%3Dyes" title="מומלץ להיכנס לחשבון, אך אין חובה לעשות זאת [o]" accesskey="o" class=""><span>כניסה לחשבון</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="אפשרויות נוספות" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="כלים אישיים" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">כלים אישיים</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="תפריט משתמש" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&amp;utm_medium=sidebar&amp;utm_campaign=C13_he.wikipedia.org&amp;uselang=he"><span>תרומה לוויקיפדיה</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A8%D7%A9%D7%9E%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&amp;returnto=%D7%A1%D7%99%D7%9E%D7%95%D7%9F+%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99&amp;returntoquery=printable%3Dyes" title="מומלץ ליצור חשבון ולהיכנס אליו, אך אין חובה לעשות זאת"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>יצירת חשבון</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%9B%D7%A0%D7%99%D7%A1%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&amp;returnto=%D7%A1%D7%99%D7%9E%D7%95%D7%9F+%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99&amp;returntoquery=printable%3Dyes" title="מומלץ להיכנס לחשבון, אך אין חובה לעשות זאת [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>כניסה לחשבון</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> דפים לעורכים שלא נכנסו לחשבון <a href="/wiki/%D7%A2%D7%96%D7%A8%D7%94:%D7%91%D7%A8%D7%95%D7%9B%D7%99%D7%9D_%D7%94%D7%91%D7%90%D7%99%D7%9D" aria-label="מידע נוסף על עריכה"><span>מידע נוסף</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%AA%D7%A8%D7%95%D7%9E%D7%95%D7%AA_%D7%A9%D7%9C%D7%99" title="רשימת העריכות שנעשו מכתובת IP זו [y]" accesskey="y"><span>תרומות</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A9%D7%99%D7%97%D7%94_%D7%A9%D7%9C%D7%99" title="דיון על העריכות שנעשו מכתובת IP זו [n]" accesskey="n"><span>שיחה</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"><div id="mw-dismissablenotice-anonplace"></div><script>(function(){var node=document.getElementById("mw-dismissablenotice-anonplace");if(node){node.outerHTML="\u003Cdiv class=\"mw-dismissable-notice\"\u003E\u003Cdiv class=\"mw-dismissable-notice-close\"\u003E[\u003Ca tabindex=\"0\" role=\"button\"\u003Eהסתרה\u003C/a\u003E]\u003C/div\u003E\u003Cdiv class=\"mw-dismissable-notice-body\"\u003E\u003C!-- CentralNotice --\u003E\u003Cdiv id=\"localNotice\" data-nosnippet=\"\"\u003E\u003Cdiv class=\"anonnotice\" lang=\"he\" dir=\"rtl\"\u003E\u003Cp\u003E\u003Cb\u003Eתמיד רציתם לכתוב בוויקיפדיה אבל לא ידעתם איך? אתם מוזמנים לסדנת עריכה בוויקיפדיה. הסדנה תתקיים בספרייה הלאומית (בבניינה החדש) בירושלים ביום שישי, 06.12.24, בשעה 09:00. להרשמה לחצו \u003Ca href=\"/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:%D7%9E%D7%99%D7%96%D7%9E%D7%99_%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94/%D7%92%D7%9C%D7%90%D7%9D/%D7%94%D7%A1%D7%A4%D7%A8%D7%99%D7%99%D7%94_%D7%94%D7%9C%D7%90%D7%95%D7%9E%D7%99%D7%AA/%D7%90%D7%99%D7%A8%D7%95%D7%A2%D7%99%D7%9D/%D7%A1%D7%93%D7%A0%D7%AA_%D7%A2%D7%A8%D7%99%D7%9B%D7%94_%D7%93%D7%A6%D7%9E%D7%91%D7%A8_2024\" title=\"ויקיפדיה:מיזמי ויקיפדיה/גלאם/הספרייה הלאומית/אירועים/סדנת עריכה דצמבר 2024\"\u003Eכאן\u003C/a\u003E.\u003C/b\u003E\n\u003C/p\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E";}}());</script></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="אתר"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="תוכן עניינים" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">תוכן עניינים</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">העברה לסרגל הצד</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">הסתרה</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">התחלה</div> </a> </li> <li id="toc-הגדרה_פורמלית" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#הגדרה_פורמלית"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>הגדרה פורמלית</span> </div> </a> <button aria-controls="toc-הגדרה_פורמלית-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>שינוי מצב התת־פרק הגדרה פורמלית</span> </button> <ul id="toc-הגדרה_פורמלית-sublist" class="vector-toc-list"> <li id="toc-הסימון_O" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#הסימון_O"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>הסימון O</span> </div> </a> <ul id="toc-הסימון_O-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-הסימון_o_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#הסימון_o_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>הסימון o</span> </div> </a> <ul id="toc-הסימון_o_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-סימונים_נוספים" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#סימונים_נוספים"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>סימונים נוספים</span> </div> </a> <ul id="toc-סימונים_נוספים-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-סיכום" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#סיכום"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>סיכום</span> </div> </a> <ul id="toc-סיכום-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-הכללה" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#הכללה"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>הכללה</span> </div> </a> <ul id="toc-הכללה-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-שימושים" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#שימושים"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>שימושים</span> </div> </a> <button aria-controls="toc-שימושים-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>שינוי מצב התת־פרק שימושים</span> </button> <ul id="toc-שימושים-sublist" class="vector-toc-list"> <li id="toc-ניתוח_סיבוכיות_אלגוריתמים" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#ניתוח_סיבוכיות_אלגוריתמים"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>ניתוח סיבוכיות אלגוריתמים</span> </div> </a> <ul id="toc-ניתוח_סיבוכיות_אלגוריתמים-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-חסם_על_קירובים" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#חסם_על_קירובים"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>חסם על קירובים</span> </div> </a> <ul id="toc-חסם_על_קירובים-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-לקריאה_נוספת" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#לקריאה_נוספת"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>לקריאה נוספת</span> </div> </a> <ul id="toc-לקריאה_נוספת-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-קישורים_חיצוניים" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#קישורים_חיצוניים"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>קישורים חיצוניים</span> </div> </a> <ul id="toc-קישורים_חיצוניים-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="תוכן עניינים" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="מצב תוכן העניינים" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">מצב תוכן העניינים</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">סימון אסימפטוטי</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="מעבר לערך בשפה אחרת. זמין ב־36 שפות" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-36" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">36 שפות</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Big_O_notation" title="Big O notation – אנגלית" lang="en" hreflang="en" data-title="Big O notation" data-language-autonym="English" data-language-local-name="אנגלית" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%85%D8%AB%D9%8A%D9%84_O_%D8%A7%D9%84%D9%83%D8%A8%D8%B1%D9%89" title="تمثيل O الكبرى – ערבית" lang="ar" hreflang="ar" data-title="تمثيل O الكبرى" data-language-autonym="العربية" data-language-local-name="ערבית" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/B%C3%B6y%C3%BCk_O_i%C5%9Far%C9%99l%C9%99r_sistemi" title="Böyük O işarələr sistemi – אזרית" lang="az" hreflang="az" data-title="Böyük O işarələr sistemi" data-language-autonym="Azərbaycanca" data-language-local-name="אזרית" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9E-%D0%BD%D0%B0%D1%82%D0%B0%D1%86%D1%8B%D1%8F" title="О-натацыя – בלארוסית" lang="be" hreflang="be" data-title="О-натацыя" data-language-autonym="Беларуская" data-language-local-name="בלארוסית" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%A1%E0%A6%BC_O_%E0%A6%B2%E0%A6%BF%E0%A6%96%E0%A6%A8%E0%A6%AA%E0%A6%A6%E0%A7%8D%E0%A6%A7%E0%A6%A4%E0%A6%BF" title="বড় O লিখনপদ্ধতি – בנגלית" lang="bn" hreflang="bn" data-title="বড় O লিখনপদ্ধতি" data-language-autonym="বাংলা" data-language-local-name="בנגלית" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Notaci%C3%B3_de_Landau" title="Notació de Landau – קטלאנית" lang="ca" hreflang="ca" data-title="Notació de Landau" data-language-autonym="Català" data-language-local-name="קטלאנית" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Landauova_notace" title="Landauova notace – צ׳כית" lang="cs" hreflang="cs" data-title="Landauova notace" data-language-autonym="Čeština" data-language-local-name="צ׳כית" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Landau-Symbole" title="Landau-Symbole – גרמנית" lang="de" hreflang="de" data-title="Landau-Symbole" data-language-autonym="Deutsch" data-language-local-name="גרמנית" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Granda_O" title="Granda O – אספרנטו" lang="eo" hreflang="eo" data-title="Granda O" data-language-autonym="Esperanto" data-language-local-name="אספרנטו" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Cota_superior_asint%C3%B3tica" title="Cota superior asintótica – ספרדית" lang="es" hreflang="es" data-title="Cota superior asintótica" data-language-autonym="Español" data-language-local-name="ספרדית" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Goi_borne_asintotiko" title="Goi borne asintotiko – בסקית" lang="eu" hreflang="eu" data-title="Goi borne asintotiko" data-language-autonym="Euskara" data-language-local-name="בסקית" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D9%85%D8%A7%D8%AF_O_%D8%A8%D8%B2%D8%B1%DA%AF" title="نماد O بزرگ – פרסית" lang="fa" hreflang="fa" data-title="نماد O بزرگ" data-language-autonym="فارسی" data-language-local-name="פרסית" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Comparaison_asymptotique" title="Comparaison asymptotique – צרפתית" lang="fr" hreflang="fr" data-title="Comparaison asymptotique" data-language-autonym="Français" data-language-local-name="צרפתית" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AC%E0%A4%A1%E0%A4%BC%E0%A4%BE_%E0%A4%93_%E0%A4%B8%E0%A4%82%E0%A4%95%E0%A5%87%E0%A4%A4%E0%A4%A8" title="बड़ा ओ संकेतन – הינדי" lang="hi" hreflang="hi" data-title="बड़ा ओ संकेतन" data-language-autonym="हिन्दी" data-language-local-name="הינדי" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/O_jel%C3%B6l%C3%A9s" title="O jelölés – הונגרית" lang="hu" hreflang="hu" data-title="O jelölés" data-language-autonym="Magyar" data-language-local-name="הונגרית" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Notasi_O_besar" title="Notasi O besar – אינדונזית" lang="id" hreflang="id" data-title="Notasi O besar" data-language-autonym="Bahasa Indonesia" data-language-local-name="אינדונזית" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/O-grande" title="O-grande – איטלקית" lang="it" hreflang="it" data-title="O-grande" data-language-autonym="Italiano" data-language-local-name="איטלקית" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%A9%E3%83%B3%E3%83%80%E3%82%A6%E3%81%AE%E8%A8%98%E5%8F%B7" title="ランダウの記号 – יפנית" lang="ja" hreflang="ja" data-title="ランダウの記号" data-language-autonym="日本語" data-language-local-name="יפנית" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%90%E1%83%A1%E1%83%98%E1%83%9B%E1%83%9E%E1%83%A2%E1%83%9D%E1%83%A2%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%90%E1%83%A6%E1%83%9C%E1%83%98%E1%83%A8%E1%83%95%E1%83%9C%E1%83%90_O-%E1%83%93%E1%83%98%E1%83%93%E1%83%98" title="ასიმპტოტური აღნიშვნა O-დიდი – גאורגית" lang="ka" hreflang="ka" data-title="ასიმპტოტური აღნიშვნა O-დიდი" data-language-autonym="ქართული" data-language-local-name="גאורגית" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%90%EA%B7%BC_%ED%91%9C%EA%B8%B0%EB%B2%95" title="점근 표기법 – קוריאנית" lang="ko" hreflang="ko" data-title="점근 표기법" data-language-autonym="한국어" data-language-local-name="קוריאנית" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Grote-O-notatie" title="Grote-O-notatie – הולנדית" lang="nl" hreflang="nl" data-title="Grote-O-notatie" data-language-autonym="Nederlands" data-language-local-name="הולנדית" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Stor_O-notasjon" title="Stor O-notasjon – נורווגית ספרותית" lang="nb" hreflang="nb" data-title="Stor O-notasjon" data-language-autonym="Norsk bokmål" data-language-local-name="נורווגית ספרותית" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Asymptotyczne_tempo_wzrostu" title="Asymptotyczne tempo wzrostu – פולנית" lang="pl" hreflang="pl" data-title="Asymptotyczne tempo wzrostu" data-language-autonym="Polski" data-language-local-name="פולנית" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Grande-O" title="Grande-O – פורטוגזית" lang="pt" hreflang="pt" data-title="Grande-O" data-language-autonym="Português" data-language-local-name="פורטוגזית" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Nota%C8%9Bia_Big_O" title="Notația Big O – רומנית" lang="ro" hreflang="ro" data-title="Notația Big O" data-language-autonym="Română" data-language-local-name="רומנית" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%C2%ABO%C2%BB_%D0%B1%D0%BE%D0%BB%D1%8C%D1%88%D0%BE%D0%B5_%D0%B8_%C2%ABo%C2%BB_%D0%BC%D0%B0%D0%BB%D0%BE%D0%B5" title="«O» большое и «o» малое – רוסית" lang="ru" hreflang="ru" data-title="«O» большое и «o» малое" data-language-autonym="Русский" data-language-local-name="רוסית" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Big_O_notation" title="Big O notation – אנגלית פשוטה" lang="en-simple" hreflang="en-simple" data-title="Big O notation" data-language-autonym="Simple English" data-language-local-name="אנגלית פשוטה" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/O_notacija" title="O notacija – סלובנית" lang="sl" hreflang="sl" data-title="O notacija" data-language-autonym="Slovenščina" data-language-local-name="סלובנית" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%92%D0%B5%D0%BB%D0%B8%D0%BA%D0%BE_%D0%9E" title="Велико О – סרבית" lang="sr" hreflang="sr" data-title="Велико О" data-language-autonym="Српски / srpski" data-language-local-name="סרבית" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Ordo" title="Ordo – שוודית" lang="sv" hreflang="sv" data-title="Ordo" data-language-autonym="Svenska" data-language-local-name="שוודית" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AA%E0%B8%B1%E0%B8%8D%E0%B8%81%E0%B8%A3%E0%B8%93%E0%B9%8C%E0%B9%82%E0%B8%AD%E0%B9%83%E0%B8%AB%E0%B8%8D%E0%B9%88" title="สัญกรณ์โอใหญ่ – תאית" lang="th" hreflang="th" data-title="สัญกรณ์โอใหญ่" data-language-autonym="ไทย" data-language-local-name="תאית" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/B%C3%BCy%C3%BCk_O_g%C3%B6sterimi" title="Büyük O gösterimi – טורקית" lang="tr" hreflang="tr" data-title="Büyük O gösterimi" data-language-autonym="Türkçe" data-language-local-name="טורקית" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D0%BE%D1%82%D0%B0%D1%86%D1%96%D1%8F_%D0%9B%D0%B0%D0%BD%D0%B4%D0%B0%D1%83" title="Нотація Ландау – אוקראינית" lang="uk" hreflang="uk" data-title="Нотація Ландау" data-language-autonym="Українська" data-language-local-name="אוקראינית" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/K%C3%BD_hi%E1%BB%87u_O_l%E1%BB%9Bn" title="Ký hiệu O lớn – וייטנאמית" lang="vi" hreflang="vi" data-title="Ký hiệu O lớn" data-language-autonym="Tiếng Việt" data-language-local-name="וייטנאמית" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%A4%A7O%E7%AC%A6%E5%8F%B7" title="大O符号 – סינית" lang="zh" hreflang="zh" data-title="大O符号" data-language-autonym="中文" data-language-local-name="סינית" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%A4%A7_O_%E7%AC%A6%E8%99%9F" title="大 O 符號 – קנטונזית" lang="yue" hreflang="yue" data-title="大 O 符號" data-language-autonym="粵語" data-language-local-name="קנטונזית" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q269878#sitelinks-wikipedia" title="עריכת קישורים בין־לשוניים" class="wbc-editpage">עריכת הקישורים</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="מרחבי שם"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99" title="צפייה בדף התוכן [c]" accesskey="c"><span>ערך</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/%D7%A9%D7%99%D7%97%D7%94:%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99" rel="discussion" title="שיחה על דף התוכן [t]" accesskey="t"><span>שיחה</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="שינוי הגוון השפה" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">עברית</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="צפיות"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99"><span>קריאה</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99&amp;action=edit" title="עריכת קוד המקור של הדף הזה [e]" accesskey="e"><span>עריכת קוד מקור</span></a></li><li id="ca-ve-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99&amp;veaction=edit" title="עריכת הדף הזה [v]" accesskey="v"><span>עריכה</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99&amp;action=history" title="גרסאות קודמות של דף זה [h]" accesskey="h"><span>גרסאות קודמות</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="כלי דף"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="כלים" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">כלים</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">כלים</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">העברה לסרגל הצד</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">הסתרה</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="אפשרויות נוספות" > <div class="vector-menu-heading"> פעולות </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99"><span>קריאה</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99&amp;action=edit" title="עריכת קוד המקור של הדף הזה [e]" accesskey="e"><span>עריכת קוד מקור</span></a></li><li id="ca-more-ve-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99&amp;veaction=edit" title="עריכת הדף הזה [v]" accesskey="v"><span>עריכה</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99&amp;action=history"><span>גרסאות קודמות</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> כללי </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%93%D7%A4%D7%99%D7%9D_%D7%94%D7%9E%D7%A7%D7%95%D7%A9%D7%A8%D7%99%D7%9D_%D7%9C%D7%9B%D7%90%D7%9F/%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99" title="רשימה של כל דפי הוויקי שמקשרים לדף הזה [j]" accesskey="j"><span>דפים המקושרים לכאן</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%A9%D7%99%D7%A0%D7%95%D7%99%D7%99%D7%9D_%D7%91%D7%93%D7%A4%D7%99%D7%9D_%D7%94%D7%9E%D7%A7%D7%95%D7%A9%D7%A8%D7%99%D7%9D/%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99" rel="nofollow" title="השינויים האחרונים בדפים המקושרים מהדף הזה [k]" accesskey="k"><span>שינויים בדפים המקושרים</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%93%D7%A4%D7%99%D7%9D_%D7%9E%D7%99%D7%95%D7%97%D7%93%D7%99%D7%9D" title="רשימה של כל הדפים המיוחדים [q]" accesskey="q"><span>דפים מיוחדים</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99&amp;oldid=39136980" title="קישור קבוע לגרסה הזאת של הדף הזה"><span>קישור קבוע</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99&amp;action=info" title="מידע נוסף על הדף הזה"><span>מידע על הדף</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%A6%D7%99%D7%98%D7%95%D7%98_%D7%93%D7%A3_%D7%96%D7%94&amp;page=%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99&amp;id=39136980&amp;wpFormIdentifier=titleform" title="מידע איך לצטט את הדף הזה"><span>ציטוט הדף הזה</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%9E%D7%A7%D7%A6%D7%A8_%D7%9B%D7%AA%D7%95%D7%91%D7%95%D7%AA&amp;url=https%3A%2F%2Fhe.wikipedia.org%2Fw%2Findex.php%3Ftitle%3D%25D7%25A1%25D7%2599%25D7%259E%25D7%2595%25D7%259F_%25D7%2590%25D7%25A1%25D7%2599%25D7%259E%25D7%25A4%25D7%2598%25D7%2595%25D7%2598%25D7%2599%26printable%3Dyes"><span>קבלת כתובת מקוצרת</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:QrCode&amp;url=https%3A%2F%2Fhe.wikipedia.org%2Fw%2Findex.php%3Ftitle%3D%25D7%25A1%25D7%2599%25D7%259E%25D7%2595%25D7%259F_%25D7%2590%25D7%25A1%25D7%2599%25D7%259E%25D7%25A4%25D7%2598%25D7%2595%25D7%2598%25D7%2599%26printable%3Dyes"><span>הורדת קוד QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> הדפסה/יצוא </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%A1%D7%A4%D7%A8&amp;bookcmd=book_creator&amp;referer=%D7%A1%D7%99%D7%9E%D7%95%D7%9F+%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99"><span>יצירת ספר</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:DownloadAsPdf&amp;page=%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99&amp;action=show-download-screen"><span>הורדה כ־PDF</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> במיזמים אחרים </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q269878" title="קישור לפריט המשויך במאגר הנתונים [g]" accesskey="g"><span>פריט ויקינתונים</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="כלי דף"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="מראה"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">מראה</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">העברה לסרגל הצד</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">הסתרה</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">מתוך ויקיפדיה, האנציקלופדיה החופשית</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="cdx-message cdx-message--block cdx-message--warning"><span class="cdx-message__icon"></span><div class="cdx-message__content">הגרסה להדפסה אינה נתמכת עוד וייתכן שיש בה שגיאות תיצוג. נא לעדכן את הסימניות בדפדפן שלך ולהשתמש בפעולת ההדפסה הרגילה של הדפדפן במקום זה.</div></div><div class="mw-content-rtl mw-parser-output" lang="he" dir="rtl"><p><b>סימון אסימפטוטי</b> (ידוע גם כ<b>סימון <a href="/wiki/%D7%90%D7%93%D7%9E%D7%95%D7%A0%D7%93_%D7%9C%D7%A0%D7%93%D7%90%D7%95" title="אדמונד לנדאו">לנדאו</a></b>) משמש ב<a href="/wiki/%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94" title="מתמטיקה">מתמטיקה</a> כ<a href="/wiki/%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%9E%D7%AA%D7%9E%D7%98%D7%99" title="סימון מתמטי">סימון</a> מקוצר שמתאר את התנהגותן של <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94" title="פונקציה">פונקציות</a> עבור ערכים הולכים וגדלים (או הולכים וקטנים), וזאת באמצעות השוואתן לפונקציות אחרות. היתרון שבשימוש בסימונים אסימפטוטיים הוא שהוא מאפשר לקבל הערכה טובה על אופן הגידול של ערכי הפונקציה מבלי שיהיה צורך לדעת אותו במדויק. לסימונים אסימפטוטיים שני תחומים מרכזיים שבהם הם יעילים: ב<a href="/wiki/%D7%9E%D7%93%D7%A2%D7%99_%D7%94%D7%9E%D7%97%D7%A9%D7%91" title="מדעי המחשב">מדעי המחשב</a> הם משמשים כדי להעריך את ה<a href="/wiki/%D7%A1%D7%99%D7%91%D7%95%D7%9B%D7%99%D7%95%D7%AA" title="סיבוכיות">סיבוכיות</a> של <a href="/wiki/%D7%90%D7%9C%D7%92%D7%95%D7%A8%D7%99%D7%AA%D7%9D" title="אלגוריתם">אלגוריתמים</a>, ובמתמטיקה הם משמשים על מנת להעריך את גודל השגיאה ב<a href="/wiki/%D7%A7%D7%99%D7%A8%D7%95%D7%91" title="קירוב">קירובים</a> שונים. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="הגדרה_פורמלית"><span id=".D7.94.D7.92.D7.93.D7.A8.D7.94_.D7.A4.D7.95.D7.A8.D7.9E.D7.9C.D7.99.D7.AA"></span>הגדרה פורמלית</h2></div> <p>השימוש בסימון אסימפטוטי מערב תמיד פונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9d93a460e6e6f85291c2df324622a50eb75661" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.697ex; height:2.009ex;" alt="{\displaystyle \ g}"></span>, על פי רוב מקבוצת <a href="/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%98%D7%91%D7%A2%D7%99" title="מספר טבעי">הטבעיים</a> לעצמה או <a href="/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%9E%D7%9E%D7%A9%D7%99" title="מספר ממשי">הממשיים</a> לעצמה. הפונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9d93a460e6e6f85291c2df324622a50eb75661" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.697ex; height:2.009ex;" alt="{\displaystyle \ g}"></span> משמשת בתור <a href="/wiki/%D7%A7%D7%A0%D7%94_%D7%9E%D7%99%D7%93%D7%94" title="קנה מידה">קנה המידה</a> ההשוואתי לפונקציות אחרות. לרוב ההתעניינות היא בקצב הגדילה של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9d93a460e6e6f85291c2df324622a50eb75661" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.697ex; height:2.009ex;" alt="{\displaystyle \ g}"></span> כאשר הערכים שהיא מקבלת <a href="/wiki/%D7%90%D7%99%D7%A0%D7%A1%D7%95%D7%A3#האינסוף_כתהליך_הגדל_כרצוננו" title="אינסוף">שואפים לאינסוף</a>, אולם הדבר אינו הכרחי. בהגדרות שנביא כאן נניח כי כל הפונקציות הן מהטבעיים לעצמם (מצב נפוץ במיוחד במדעי המחשב, מכיוון שמדדי <a href="/wiki/%D7%A1%D7%99%D7%91%D7%95%D7%9B%D7%99%D7%95%D7%AA_%D7%96%D7%9E%D7%9F" title="סיבוכיות זמן">סיבוכיות זמן</a> וזיכרון הם חיוביים ובדידים). עם זאת, גם ללא ההנחה נדרוש מ-g להיות אי שלילית החל מאיבר מסוים. </p> <div class="mw-heading mw-heading3"><h3 id="הסימון_O"><span id=".D7.94.D7.A1.D7.99.D7.9E.D7.95.D7.9F_O"></span>הסימון O</h3></div> <figure typeof="mw:File/Thumb"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5:Big-O-notation.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Big-O-notation.png/250px-Big-O-notation.png" decoding="async" width="250" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Big-O-notation.png/375px-Big-O-notation.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/89/Big-O-notation.png/500px-Big-O-notation.png 2x" data-file-width="661" data-file-height="622" /></a><figcaption>דוגמה לסימון O גדול: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\in O(g(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>&#x2208;<!-- ∈ --></mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\in O(g(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c629729344c601e5273720b8e0f33a18e6338d3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.096ex; height:2.843ex;" alt="{\displaystyle f(x)\in O(g(x))}"></span> כי קיים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c&gt;0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>&gt;</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c&gt;0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ba126f626d61752f62eaacaf11761a54de4dc84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c&gt;0}"></span> (ובפרט <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e3467f9e219a5ea38a30da5c3a02c2c23f61a79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c=1}"></span>) ו-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span> (ובפרט <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}=5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fb111f85ebf1abbc8b78932a621d47dab364bda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.645ex; height:2.509ex;" alt="{\displaystyle x_{0}=5}"></span>) עבורם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(n)\leq \;c\cdot g(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>&#x2264;<!-- ≤ --></mo> <mspace width="thickmathspace" /> <mi>c</mi> <mo>&#x22C5;<!-- ⋅ --></mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(n)\leq \;c\cdot g(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3711e74380fab4106b9a9979d906d3b2d7bfcb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.813ex; height:2.843ex;" alt="{\displaystyle \ f(n)\leq \;c\cdot g(n)}"></span> לכל <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\geq x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>&#x2265;<!-- ≥ --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\geq x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06eb265a902eeabdb1149c4e0682a9a52ec3a4a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.812ex; height:2.343ex;" alt="{\displaystyle x\geq x_{0}}"></span>.</figcaption></figure> <p>הסימון הנפוץ ביותר הוא הסימון "O" (קרי: אוֹוּ גדול). כאשר עוסקים בשאיפה של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ g(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ g(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9040eebfc62420cc4bf427ca04e15c95941ee0b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.901ex; height:2.843ex;" alt="{\displaystyle \ g(n)}"></span> לאינסוף, מוגדרת הקבוצה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ O(g(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ O(g(n))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6e9dee2120799b335ff0c2ed3af465acffdff81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.483ex; height:2.843ex;" alt="{\displaystyle \ O(g(n))}"></span> בתור אוסף כל הפונקציות <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f35eed34cace2fbdf1e38bcbc8c81db3b21b2db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.063ex; height:2.843ex;" alt="{\displaystyle \ f(n)}"></span> (בעלות אותו <a href="/wiki/%D7%AA%D7%97%D7%95%D7%9D_%D7%A9%D7%9C_%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94" title="תחום של פונקציה">תחום</a> ו<a href="/wiki/%D7%98%D7%95%D7%95%D7%97_%D7%A9%D7%9C_%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94" title="טווח של פונקציה">טווח</a> כמו <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9d93a460e6e6f85291c2df324622a50eb75661" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.697ex; height:2.009ex;" alt="{\displaystyle \ g}"></span>) עבורן קיימים קבועים חיוביים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ n_{0}\in \mathbb {N} ,0&lt;c\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>&#x2208;<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>,</mo> <mn>0</mn> <mo>&lt;</mo> <mi>c</mi> <mo>&#x2208;<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ n_{0}\in \mathbb {N} ,0&lt;c\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed355a899d764a1cb1063100efe70e55a91eaa37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.369ex; height:2.509ex;" alt="{\displaystyle \ n_{0}\in \mathbb {N} ,0&lt;c\in \mathbb {R} }"></span> כך שלכל <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ n&gt;n_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>n</mi> <mo>&gt;</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ n&gt;n_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/264bbee02489da5968c49b7d6f720dfa2b13202b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.523ex; height:2.176ex;" alt="{\displaystyle \ n&gt;n_{0}}"></span> מתקיים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(n)\leq \;c\cdot g(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>&#x2264;<!-- ≤ --></mo> <mspace width="thickmathspace" /> <mi>c</mi> <mo>&#x22C5;<!-- ⋅ --></mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(n)\leq \;c\cdot g(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3711e74380fab4106b9a9979d906d3b2d7bfcb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.813ex; height:2.843ex;" alt="{\displaystyle \ f(n)\leq \;c\cdot g(n)}"></span>. כלומר: עבור ערכי n הולכים וגדלים שמקבלת <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5ff7312a01506eee6ecea7dca662763a101c9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.859ex; height:2.509ex;" alt="{\displaystyle \ f}"></span>, היא קטנה יותר מ-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9d93a460e6e6f85291c2df324622a50eb75661" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.697ex; height:2.009ex;" alt="{\displaystyle \ g}"></span> עד כדי כפל בקבוע. </p><p>בסימון מתמטי המשתמש ב<a href="/wiki/%D7%92%D7%91%D7%95%D7%9C_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="גבול (מתמטיקה)">גבולות</a>, ניתן להגדיר כי פונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f35eed34cace2fbdf1e38bcbc8c81db3b21b2db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.063ex; height:2.843ex;" alt="{\displaystyle \ f(n)}"></span> שייכת ל-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ O(g(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ O(g(n))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6e9dee2120799b335ff0c2ed3af465acffdff81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.483ex; height:2.843ex;" alt="{\displaystyle \ O(g(n))}"></span> אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \limsup _{n\to \infty }{\frac {f(n)}{g(n)}}&lt;\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <munder> <mo movablelimits="true" form="prefix">lim&#x2006;sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> <mi mathvariant="normal">&#x221E;<!-- ∞ --></mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>&lt;</mo> <mi mathvariant="normal">&#x221E;<!-- ∞ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \limsup _{n\to \infty }{\frac {f(n)}{g(n)}}&lt;\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22970a1efd142512584c25219fc9ffa490339a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.215ex; height:6.509ex;" alt="{\displaystyle \ \limsup _{n\to \infty }{\frac {f(n)}{g(n)}}&lt;\infty }"></span>. שתי ההגדרות שקולות. </p><p>מטעמי פשטות, נהוג לכתוב <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(n)=O(g(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(n)=O(g(n))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69def620fdfe7d6af8d0a1dbd081f92b564f22b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.064ex; height:2.843ex;" alt="{\displaystyle \ f(n)=O(g(n))}"></span> כאשר הכוונה היא שמתקיים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(n)\in O(g(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>&#x2208;<!-- ∈ --></mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(n)\in O(g(n))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae7c4f25022911a3da32b4265255223ae826421b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.807ex; height:2.843ex;" alt="{\displaystyle \ f(n)\in O(g(n))}"></span>. <a href="/wiki/%D7%91%D7%99%D7%98%D7%95%D7%99_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="ביטוי (מתמטיקה)">ביטוי</a> זה עשוי לגרום לבלבול אם מתייחסים אליו כמציין שוויון ולא שייכות; בפרט, הוא אינו סימטרי. למשל, מתקיים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ n=O(n^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>n</mi> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ n=O(n^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f758f90f9dd1c942596c2fe263d99dc7138af3cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.105ex; height:3.176ex;" alt="{\displaystyle \ n=O(n^{2})}"></span>, אבל <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ O(n^{2})\neq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>&#x2260;<!-- ≠ --></mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ O(n^{2})\neq n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81e21adba1cf252932c16650d7882892f88c5b21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.105ex; height:3.176ex;" alt="{\displaystyle \ O(n^{2})\neq n}"></span>. </p><p><span id="או_קטן" class="citation"></span> </p> <div class="mw-heading mw-heading3"><h3 id="הסימון_o_2"><span id=".D7.94.D7.A1.D7.99.D7.9E.D7.95.D7.9F_o_2"></span>הסימון o</h3></div> <p>בעוד הסימון O מציין כי הפונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5ff7312a01506eee6ecea7dca662763a101c9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.859ex; height:2.509ex;" alt="{\displaystyle \ f}"></span> חסומה בקצב גידולה על ידי קצב הגידול של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9d93a460e6e6f85291c2df324622a50eb75661" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.697ex; height:2.009ex;" alt="{\displaystyle \ g}"></span>, הרי שהסימון "o" (קרי: אוֹוּ קטן) בא לציין כי קצב הגידול של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5ff7312a01506eee6ecea7dca662763a101c9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.859ex; height:2.509ex;" alt="{\displaystyle \ f}"></span> קטן <b>ממש</b> יחסית לקצב הגידול של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9d93a460e6e6f85291c2df324622a50eb75661" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.697ex; height:2.009ex;" alt="{\displaystyle \ g}"></span>. </p><p>בצורה פורמלית, אומרים ש-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f35eed34cace2fbdf1e38bcbc8c81db3b21b2db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.063ex; height:2.843ex;" alt="{\displaystyle \ f(n)}"></span> היא <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ o(g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>o</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ o(g)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/852ec1553c97e226e2926bc77d3b38214d2f8d9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.634ex; height:2.843ex;" alt="{\displaystyle \ o(g)}"></span> אם <b>לכל</b> קבוע <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73f25683198a37f6e98e40127297a5e4d87ffeec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.587ex; height:1.676ex;" alt="{\displaystyle \ c}"></span>חיובי קיים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ n_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ n_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bc307798f3c16c793d1bab3637270c410067a89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.03ex; height:2.009ex;" alt="{\displaystyle \ n_{0}}"></span> כך שלכל <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ n&gt;n_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>n</mi> <mo>&gt;</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ n&gt;n_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/264bbee02489da5968c49b7d6f720dfa2b13202b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.523ex; height:2.176ex;" alt="{\displaystyle \ n&gt;n_{0}}"></span> מתקיים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(n)&lt;c\cdot g(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>&lt;</mo> <mi>c</mi> <mo>&#x22C5;<!-- ⋅ --></mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(n)&lt;c\cdot g(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f73d2f8dfa83a9b98d1455d180af7c82465267a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.168ex; height:2.843ex;" alt="{\displaystyle \ f(n)&lt;c\cdot g(n)}"></span>. בסימון באמצעות גבולות ניתן לתאר תכונה זו על ידי הגבול <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> <mi mathvariant="normal">&#x221E;<!-- ∞ --></mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/653d32a24fd2eeed5e05a9493f3ef17eaeeae666" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.207ex; height:6.509ex;" alt="{\displaystyle \ \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=0}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="סימונים_נוספים"><span id=".D7.A1.D7.99.D7.9E.D7.95.D7.A0.D7.99.D7.9D_.D7.A0.D7.95.D7.A1.D7.A4.D7.99.D7.9D"></span>סימונים נוספים</h3></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=\Omega (g)\equiv \exists 0&lt;c\in \mathbb {R} ,\;\exists n_{0}\in \mathbb {N} \;s.t\;\forall n_{0}&lt;n\in \mathbb {N} \;c\cdot g(n)\leq f(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mi mathvariant="normal">&#x03A9;<!-- Ω --></mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>&#x2261;<!-- ≡ --></mo> <mi mathvariant="normal">&#x2203;<!-- ∃ --></mi> <mn>0</mn> <mo>&lt;</mo> <mi>c</mi> <mo>&#x2208;<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">&#x2203;<!-- ∃ --></mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>&#x2208;<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mspace width="thickmathspace" /> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mspace width="thickmathspace" /> <mi mathvariant="normal">&#x2200;<!-- ∀ --></mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>&lt;</mo> <mi>n</mi> <mo>&#x2208;<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mspace width="thickmathspace" /> <mi>c</mi> <mo>&#x22C5;<!-- ⋅ --></mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>&#x2264;<!-- ≤ --></mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=\Omega (g)\equiv \exists 0&lt;c\in \mathbb {R} ,\;\exists n_{0}\in \mathbb {N} \;s.t\;\forall n_{0}&lt;n\in \mathbb {N} \;c\cdot g(n)\leq f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20405c7564148662268fc5e823956c10f2b75dc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:65.337ex; height:2.843ex;" alt="{\displaystyle f=\Omega (g)\equiv \exists 0&lt;c\in \mathbb {R} ,\;\exists n_{0}\in \mathbb {N} \;s.t\;\forall n_{0}&lt;n\in \mathbb {N} \;c\cdot g(n)\leq f(n)}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=\omega (g)\equiv \forall 0&lt;c\in \mathbb {R} ,\;\exists n_{0}\in \mathbb {N} \;s.t\;\forall n_{0}&lt;n\in \mathbb {N} \;c\cdot g(n)&lt;f(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mi>&#x03C9;<!-- ω --></mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>&#x2261;<!-- ≡ --></mo> <mi mathvariant="normal">&#x2200;<!-- ∀ --></mi> <mn>0</mn> <mo>&lt;</mo> <mi>c</mi> <mo>&#x2208;<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">&#x2203;<!-- ∃ --></mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>&#x2208;<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mspace width="thickmathspace" /> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mspace width="thickmathspace" /> <mi mathvariant="normal">&#x2200;<!-- ∀ --></mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>&lt;</mo> <mi>n</mi> <mo>&#x2208;<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mspace width="thickmathspace" /> <mi>c</mi> <mo>&#x22C5;<!-- ⋅ --></mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>&lt;</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=\omega (g)\equiv \forall 0&lt;c\in \mathbb {R} ,\;\exists n_{0}\in \mathbb {N} \;s.t\;\forall n_{0}&lt;n\in \mathbb {N} \;c\cdot g(n)&lt;f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e465e7032e0f615120d4ca76d83852d7a62544b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:65.105ex; height:2.843ex;" alt="{\displaystyle f=\omega (g)\equiv \forall 0&lt;c\in \mathbb {R} ,\;\exists n_{0}\in \mathbb {N} \;s.t\;\forall n_{0}&lt;n\in \mathbb {N} \;c\cdot g(n)&lt;f(n)}"></span> </p><p>באמצעות הסימונים שהוגדרו לעיל נהוג להגדיר סימון נוסף: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f=\Theta (g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> <mo>=</mo> <mi mathvariant="normal">&#x0398;<!-- Θ --></mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f=\Theta (g)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0be8183f731b175f59389222d1968154604adeb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.691ex; height:2.843ex;" alt="{\displaystyle \ f=\Theta (g)}"></span> אם ורק אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f=O(g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f=O(g)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/251e03aa43a511f24d4dbef85ddb94bfebec3d41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.656ex; height:2.843ex;" alt="{\displaystyle \ f=O(g)}"></span> וגם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f=\Omega (g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> <mo>=</mo> <mi mathvariant="normal">&#x03A9;<!-- Ω --></mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f=\Omega (g)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/037d22a1a207c3f6f013a8a787635fdd269a8a0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.561ex; height:2.843ex;" alt="{\displaystyle \ f=\Omega (g)}"></span>, אולם לעיתים מגדירים כי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=\Theta (g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mi mathvariant="normal">&#x0398;<!-- Θ --></mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=\Theta (g)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e324e9a916a2573356e5bbe83ecffe52168b9d2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.11ex; height:2.843ex;" alt="{\displaystyle f=\Theta (g)}"></span> אם ורק אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists c_{1},c_{2}&gt;0,\exists n_{0}\in \mathbb {N} \;s.t\;\forall n_{0}\leq n\;c_{1}\cdot g(n)\leq f(n)\leq c_{2}\cdot g(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&#x2203;<!-- ∃ --></mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>&gt;</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">&#x2203;<!-- ∃ --></mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>&#x2208;<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mspace width="thickmathspace" /> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mspace width="thickmathspace" /> <mi mathvariant="normal">&#x2200;<!-- ∀ --></mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>&#x2264;<!-- ≤ --></mo> <mi>n</mi> <mspace width="thickmathspace" /> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>&#x22C5;<!-- ⋅ --></mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>&#x2264;<!-- ≤ --></mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>&#x2264;<!-- ≤ --></mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>&#x22C5;<!-- ⋅ --></mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists c_{1},c_{2}&gt;0,\exists n_{0}\in \mathbb {N} \;s.t\;\forall n_{0}\leq n\;c_{1}\cdot g(n)\leq f(n)\leq c_{2}\cdot g(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a320f031c89481fa4330a0cae439c28eabc4365b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:59.938ex; height:2.843ex;" alt="{\displaystyle \exists c_{1},c_{2}&gt;0,\exists n_{0}\in \mathbb {N} \;s.t\;\forall n_{0}\leq n\;c_{1}\cdot g(n)\leq f(n)\leq c_{2}\cdot g(n)}"></span>, ולא באמצעות התנאי לעיל, למרות השקילות.</li></ul> <p>פירוש הסימון <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f=\Omega (g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> <mo>=</mo> <mi mathvariant="normal">&#x03A9;<!-- Ω --></mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f=\Omega (g)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/037d22a1a207c3f6f013a8a787635fdd269a8a0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.561ex; height:2.843ex;" alt="{\displaystyle \ f=\Omega (g)}"></span> הוא ש-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5ff7312a01506eee6ecea7dca662763a101c9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.859ex; height:2.509ex;" alt="{\displaystyle \ f}"></span> גדל לפחות בקצב של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9d93a460e6e6f85291c2df324622a50eb75661" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.697ex; height:2.009ex;" alt="{\displaystyle \ g}"></span>. פירוש הסימון <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f=\omega (g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> <mo>=</mo> <mi>&#x03C9;<!-- ω --></mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f=\omega (g)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85c90ca181f23aef525f0c3906924b84667985fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.329ex; height:2.843ex;" alt="{\displaystyle \ f=\omega (g)}"></span> הוא ש-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5ff7312a01506eee6ecea7dca662763a101c9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.859ex; height:2.509ex;" alt="{\displaystyle \ f}"></span> גדל בקצב שגדול ממש מהקצב של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9d93a460e6e6f85291c2df324622a50eb75661" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.697ex; height:2.009ex;" alt="{\displaystyle \ g}"></span>, ופירוש הסימון <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f=\Theta (g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> <mo>=</mo> <mi mathvariant="normal">&#x0398;<!-- Θ --></mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f=\Theta (g)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0be8183f731b175f59389222d1968154604adeb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.691ex; height:2.843ex;" alt="{\displaystyle \ f=\Theta (g)}"></span> הוא שקצבי הגידול של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5ff7312a01506eee6ecea7dca662763a101c9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.859ex; height:2.509ex;" alt="{\displaystyle \ f}"></span> ו-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9d93a460e6e6f85291c2df324622a50eb75661" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.697ex; height:2.009ex;" alt="{\displaystyle \ g}"></span> הם שווים אסימפטוטית (כלומר, שתי הפונקציות הן מאותו <a href="/wiki/%D7%A1%D7%93%D7%A8_%D7%92%D7%95%D7%93%D7%9C" title="סדר גודל">סדר גודל</a>). </p> <div class="mw-heading mw-heading3"><h3 id="סיכום"><span id=".D7.A1.D7.99.D7.9B.D7.95.D7.9D"></span>סיכום</h3></div> <p>סיכום חמשת הסימונים האסימפטוטיים המקובלים מוצג בטבלה הבאה: </p> <table border="1" cellpadding="4" cellspacing="0"> <tbody><tr> <th>סימון </th> <th>הגדרה </th> <th>הגדרה מתמטית </th></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)\in O(g(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>&#x2208;<!-- ∈ --></mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)\in O(g(n))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c13d888fa2979a7681ad477071a84645f8fc3e43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.226ex; height:2.843ex;" alt="{\displaystyle f(n)\in O(g(n))}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4ad18070e494503403daf39398e711c1378348e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.32ex; height:2.843ex;" alt="{\displaystyle g(n)}"></span> חסם עליון אסימפטוטי </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \limsup _{n\to \infty }\left|{\frac {f(n)}{g(n)}}\right|&lt;\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim&#x2006;sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> <mi mathvariant="normal">&#x221E;<!-- ∞ --></mi> </mrow> </munder> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mo>&lt;</mo> <mi mathvariant="normal">&#x221E;<!-- ∞ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \limsup _{n\to \infty }\left|{\frac {f(n)}{g(n)}}\right|&lt;\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aac0e75907cefab6fffa07f86b6384da5367ca6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.54ex; height:6.509ex;" alt="{\displaystyle \limsup _{n\to \infty }\left|{\frac {f(n)}{g(n)}}\right|&lt;\infty }"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)\in o(g(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>&#x2208;<!-- ∈ --></mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)\in o(g(n))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/540b11c5bbe6be7d47348bdac0ddeaa026eb5903" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.58ex; height:2.843ex;" alt="{\displaystyle f(n)\in o(g(n))}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4ad18070e494503403daf39398e711c1378348e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.32ex; height:2.843ex;" alt="{\displaystyle g(n)}"></span> שולט אסימפטוטית </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> <mi mathvariant="normal">&#x221E;<!-- ∞ --></mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/492b43a40ba041a1f958d4e3923eed6fd947f08c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.239ex; height:6.509ex;" alt="{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=0}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)\in \Omega (g(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>&#x2208;<!-- ∈ --></mo> <mi mathvariant="normal">&#x03A9;<!-- Ω --></mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)\in \Omega (g(n))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/036d8a5b95067b85b075fe077648422a649271e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.131ex; height:2.843ex;" alt="{\displaystyle f(n)\in \Omega (g(n))}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4ad18070e494503403daf39398e711c1378348e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.32ex; height:2.843ex;" alt="{\displaystyle g(n)}"></span> חסם תחתון אסימפטוטי </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \liminf _{n\to \infty }\left|{\frac {f(n)}{g(n)}}\right|&gt;0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim&#x2006;inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> <mi mathvariant="normal">&#x221E;<!-- ∞ --></mi> </mrow> </munder> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mo>&gt;</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \liminf _{n\to \infty }\left|{\frac {f(n)}{g(n)}}\right|&gt;0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/facca8c7a0131b37513f9ffbd5457fc0addaade3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.529ex; height:6.509ex;" alt="{\displaystyle \liminf _{n\to \infty }\left|{\frac {f(n)}{g(n)}}\right|&gt;0}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)\in \omega (g(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>&#x2208;<!-- ∈ --></mo> <mi>&#x03C9;<!-- ω --></mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)\in \omega (g(n))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a52029c64a885b24fba54fc0513c6ec54c8be42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.898ex; height:2.843ex;" alt="{\displaystyle f(n)\in \omega (g(n))}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4ad18070e494503403daf39398e711c1378348e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.32ex; height:2.843ex;" alt="{\displaystyle g(n)}"></span> זניח אסימפטוטית </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> <mi mathvariant="normal">&#x221E;<!-- ∞ --></mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mi mathvariant="normal">&#x221E;<!-- ∞ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94f1a706b1f85b13d89bb38ef7a1486a755caa3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.401ex; height:6.509ex;" alt="{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=\infty }"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)\in \Theta (g(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>&#x2208;<!-- ∈ --></mo> <mi mathvariant="normal">&#x0398;<!-- Θ --></mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)\in \Theta (g(n))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09e673203d32ca73205115ec3e2e80e918c01905" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.261ex; height:2.843ex;" alt="{\displaystyle f(n)\in \Theta (g(n))}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4ad18070e494503403daf39398e711c1378348e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.32ex; height:2.843ex;" alt="{\displaystyle g(n)}"></span> חסם הדוק אסימפטוטית </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0&lt;\liminf _{n\to \infty }\left|{\frac {f(n)}{g(n)}}\right|\leq \limsup _{n\to \infty }\left|{\frac {f(n)}{g(n)}}\right|&lt;\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>&lt;</mo> <munder> <mo movablelimits="true" form="prefix">lim&#x2006;inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> <mi mathvariant="normal">&#x221E;<!-- ∞ --></mi> </mrow> </munder> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mo>&#x2264;<!-- ≤ --></mo> <munder> <mo movablelimits="true" form="prefix">lim&#x2006;sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> <mi mathvariant="normal">&#x221E;<!-- ∞ --></mi> </mrow> </munder> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mo>&lt;</mo> <mi mathvariant="normal">&#x221E;<!-- ∞ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0&lt;\liminf _{n\to \infty }\left|{\frac {f(n)}{g(n)}}\right|\leq \limsup _{n\to \infty }\left|{\frac {f(n)}{g(n)}}\right|&lt;\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2424d06bb97918c630b723265b3a266a64bd764" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.168ex; height:6.509ex;" alt="{\displaystyle 0&lt;\liminf _{n\to \infty }\left|{\frac {f(n)}{g(n)}}\right|\leq \limsup _{n\to \infty }\left|{\frac {f(n)}{g(n)}}\right|&lt;\infty }"></span> </td></tr></tbody></table> <p>כאשר הכוונה בכך שגבול כלשהו קטן מאינסוף היא שהגבול הוא מספר ממשי. </p> <div class="mw-heading mw-heading2"><h2 id="הכללה"><span id=".D7.94.D7.9B.D7.9C.D7.9C.D7.94"></span>הכללה</h2></div> <p>אף שהחסמים שהוצגו לעיל עוסקים בפונקציות מהמספרים הטבעיים לעצמם ובתכונותיהן כאשר הן מקבלות ערכים השואפים לאינסוף, ניתן להכליל את הסימון לטיפול במקרים אחרים. עיקר החשיבות בביצוע הכללה שכזו הוא בהבהרה של הגודל אליו שואפים הערכים שהפונקציות מקבלות. </p><p>לדוגמה, עבור פונקציה ממשיות <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69df108372019d93cfdc04fabe9dbb4cd67e4d59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.998ex; height:2.843ex;" alt="{\displaystyle \ f(x)}"></span> ופונקציה חיובית <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \phi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>&#x03D5;<!-- ϕ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \phi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/318bc365200b1f6bd92612e4f358d25398515846" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.105ex; height:2.843ex;" alt="{\displaystyle \ \phi (x)}"></span> אומרים ש-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(x)=O(\phi (x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>&#x03D5;<!-- ϕ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(x)=O(\phi (x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4818d1efc852fdcc087f223efa5e387c537d14f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.204ex; height:2.843ex;" alt="{\displaystyle \ f(x)=O(\phi (x))}"></span> כאשר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x\to x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>x</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x\to x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e6eb49ce3865e50e2c70a58f627136ec8dfc14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.908ex; height:2.176ex;" alt="{\displaystyle \ x\to x_{0}}"></span> אם קיימים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ c&gt;0,\delta &gt;0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>c</mi> <mo>&gt;</mo> <mn>0</mn> <mo>,</mo> <mi>&#x03B4;<!-- δ --></mi> <mo>&gt;</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ c&gt;0,\delta &gt;0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31988f8703f38bdb1401c6dbcc4a3096f62fdae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.192ex; height:2.676ex;" alt="{\displaystyle \ c&gt;0,\delta &gt;0}"></span> כך שלכל x בסביבה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{\delta }(x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03B4;<!-- δ --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{\delta }(x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9198e515dd00d4692e93cfb030ef5205a9d3e927" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.033ex; height:2.843ex;" alt="{\displaystyle N_{\delta }(x_{0})}"></span> מתקיים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ |f(x)|\leq c\cdot \phi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>&#x2264;<!-- ≤ --></mo> <mi>c</mi> <mo>&#x22C5;<!-- ⋅ --></mo> <mi>&#x03D5;<!-- ϕ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ |f(x)|\leq c\cdot \phi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/677aa94c73c3ba2bda6643adbe518564d38b343b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.601ex; height:2.843ex;" alt="{\displaystyle \ |f(x)|\leq c\cdot \phi (x)}"></span>. כמו כן, אומרים ש-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(x)=o(\phi (x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>&#x03D5;<!-- ϕ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(x)=o(\phi (x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/814854a6ab82bad3a5dc7125b198ee4d912fa8b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.558ex; height:2.843ex;" alt="{\displaystyle \ f(x)=o(\phi (x))}"></span> כאשר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x\to x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>x</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x\to x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e6eb49ce3865e50e2c70a58f627136ec8dfc14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.908ex; height:2.176ex;" alt="{\displaystyle \ x\to x_{0}}"></span> אם לכל <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon &gt;0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>&#x03F5;<!-- ϵ --></mi> <mo>&gt;</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon &gt;0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon &gt;0}"></span> קיים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta &gt;0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>&#x03B4;<!-- δ --></mi> <mo>&gt;</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta &gt;0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/595d5cea06fdcaf2642caf549eda2cfc537958a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta &gt;0}"></span> כך שלכל x בסביבה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{\delta }(x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03B4;<!-- δ --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{\delta }(x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9198e515dd00d4692e93cfb030ef5205a9d3e927" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.033ex; height:2.843ex;" alt="{\displaystyle N_{\delta }(x_{0})}"></span> מתקיים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ |f(x)|\leq \epsilon \cdot \phi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>&#x2264;<!-- ≤ --></mo> <mi>&#x03F5;<!-- ϵ --></mi> <mo>&#x22C5;<!-- ⋅ --></mo> <mi>&#x03D5;<!-- ϕ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ |f(x)|\leq \epsilon \cdot \phi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0f9c7c83b719cec7f460b66ba0930b701aa2c2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.538ex; height:2.843ex;" alt="{\displaystyle \ |f(x)|\leq \epsilon \cdot \phi (x)}"></span>.התנאי האחרון שקול לדרישה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \lim _{x\to x_{0}}{\frac {f(x)}{\phi (x)}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>&#x03D5;<!-- ϕ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \lim _{x\to x_{0}}{\frac {f(x)}{\phi (x)}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fe3a348139fb5439780f8db4614d93d163620b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.331ex; height:6.509ex;" alt="{\displaystyle \ \lim _{x\to x_{0}}{\frac {f(x)}{\phi (x)}}=0}"></span>. באופן דומה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(x)=o(\phi (x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>&#x03D5;<!-- ϕ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(x)=o(\phi (x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/814854a6ab82bad3a5dc7125b198ee4d912fa8b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.558ex; height:2.843ex;" alt="{\displaystyle \ f(x)=o(\phi (x))}"></span> כאשר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>x</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> <mi mathvariant="normal">&#x221E;<!-- ∞ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63ffb453568dec1ac87c67ddd5b7d416f7c31386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.848ex; height:1.843ex;" alt="{\displaystyle \ x\to \infty }"></span> אם ורק אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \lim _{x\to \infty }{\frac {f(x)}{\phi (x)}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> <mi mathvariant="normal">&#x221E;<!-- ∞ --></mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>&#x03D5;<!-- ϕ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \lim _{x\to \infty }{\frac {f(x)}{\phi (x)}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0cd3bad72a5fccfe300d1fc245c5064a43533aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.203ex; height:6.509ex;" alt="{\displaystyle \ \lim _{x\to \infty }{\frac {f(x)}{\phi (x)}}=0}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="שימושים"><span id=".D7.A9.D7.99.D7.9E.D7.95.D7.A9.D7.99.D7.9D"></span>שימושים</h2></div> <div class="mw-heading mw-heading3"><h3 id="ניתוח_סיבוכיות_אלגוריתמים"><span id=".D7.A0.D7.99.D7.AA.D7.95.D7.97_.D7.A1.D7.99.D7.91.D7.95.D7.9B.D7.99.D7.95.D7.AA_.D7.90.D7.9C.D7.92.D7.95.D7.A8.D7.99.D7.AA.D7.9E.D7.99.D7.9D"></span>ניתוח סיבוכיות אלגוריתמים</h3></div> <p>נניח כי אלגוריתם <a href="/wiki/%D7%9E%D7%99%D7%95%D7%9F_(%D7%9E%D7%93%D7%A2%D7%99_%D7%94%D7%9E%D7%97%D7%A9%D7%91)" class="mw-redirect" title="מיון (מדעי המחשב)">מיון</a> מסוים מבצע בדיוק <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ 4n^{2}-2n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mn>4</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>&#x2212;<!-- − --></mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 4n^{2}-2n+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/581e2781cd727f5137629161c13a38c645b19ac6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.592ex; height:2.843ex;" alt="{\displaystyle \ 4n^{2}-2n+1}"></span> פעולות בסיסיות על קלט מגודל <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaf8b0f621a23f81aa20d63b5cd59d3dcad83ccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.975ex; height:1.676ex;" alt="{\displaystyle \ n}"></span>. ככל ש-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaf8b0f621a23f81aa20d63b5cd59d3dcad83ccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.975ex; height:1.676ex;" alt="{\displaystyle \ n}"></span> גדל כך ההשפעה של הגורם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4397d73fbceff200bcd50ab68a0bfb749460e3f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.03ex; height:2.676ex;" alt="{\displaystyle \ n^{2}}"></span> גדלה בעוד ההשפעה של שאר הגורמים הופכת לזניחה. לדוגמה, עבור <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ n=500}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>n</mi> <mo>=</mo> <mn>500</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ n=500}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4641c78801e4002bbce8fdebc9bc7508095f7ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.561ex; height:2.176ex;" alt="{\displaystyle \ n=500}"></span>, גודלו של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ 4n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mn>4</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 4n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c53a0f08432b6cc52ba948d6e9aa385cdb62a2f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.192ex; height:2.676ex;" alt="{\displaystyle \ 4n^{2}}"></span> גדול פי 1,000 מגודלו של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ 2n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mn>2</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 2n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe73c1158ef6b049b32da2687b756dbf79449982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.138ex; height:2.176ex;" alt="{\displaystyle \ 2n}"></span>, ופי מיליון מגודלו של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c043461e59fa0e8d37d8acbd6d7e5b782dd84246" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \ 1}"></span>, ולכן שני הגורמים הללו זניחים ביחס ל-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ 4n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mn>4</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 4n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c53a0f08432b6cc52ba948d6e9aa385cdb62a2f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.192ex; height:2.676ex;" alt="{\displaystyle \ 4n^{2}}"></span> ובמרבית השימושים לא תהיה להם כל חשיבות. </p><p>בנוסף, למקדם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aa8e394ea29e1ced6a36fe8a7d3c9bccd89b8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \ 4}"></span> של הביטוי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ 4n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mn>4</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 4n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c53a0f08432b6cc52ba948d6e9aa385cdb62a2f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.192ex; height:2.676ex;" alt="{\displaystyle \ 4n^{2}}"></span> יש חשיבות אפסית בלבד כאשר משווים את הביטוי לביטויים גדולים יותר. למשל, אם משווים את <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ 4n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mn>4</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 4n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c53a0f08432b6cc52ba948d6e9aa385cdb62a2f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.192ex; height:2.676ex;" alt="{\displaystyle \ 4n^{2}}"></span> ל-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ n^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ n^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/166e76b10bd53afce9bc185ca930c8ed01c1ed3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.03ex; height:2.676ex;" alt="{\displaystyle \ n^{3}}"></span>, כבר עבור <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ n=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>n</mi> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ n=5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1be49301a37c28dc4dd72b78001bee663c05e43a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.236ex; height:2.176ex;" alt="{\displaystyle \ n=5}"></span> הביטוי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ n^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ n^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/166e76b10bd53afce9bc185ca930c8ed01c1ed3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.03ex; height:2.676ex;" alt="{\displaystyle \ n^{3}}"></span> יהיה גדול יותר. אפילו אם המקדם של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4397d73fbceff200bcd50ab68a0bfb749460e3f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.03ex; height:2.676ex;" alt="{\displaystyle \ n^{2}}"></span> יהיה 1,000,000, עבור <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ n=1,000,000}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>000</mn> <mo>,</mo> <mn>000</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ n=1,000,000}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3b7bd7499afd0ae6dacd3c2b3d345690df70c7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.279ex; height:2.509ex;" alt="{\displaystyle \ n=1,000,000}"></span> ומעלה שוב יתקבל כי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ n^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ n^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/166e76b10bd53afce9bc185ca930c8ed01c1ed3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.03ex; height:2.676ex;" alt="{\displaystyle \ n^{3}}"></span> גדול יותר. על כן, כאשר מתעניינים בגידול האסימפטוטי של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4397d73fbceff200bcd50ab68a0bfb749460e3f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.03ex; height:2.676ex;" alt="{\displaystyle \ n^{2}}"></span> אין חשיבות למקדם שלו. </p><p>בשל כך משתמשים בסימון <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ 4n^{2}-2n+1=O(n^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mn>4</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>&#x2212;<!-- − --></mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 4n^{2}-2n+1=O(n^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f2ad25610a6012481d7250d30e95198e4ca34b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.722ex; height:3.176ex;" alt="{\displaystyle \ 4n^{2}-2n+1=O(n^{2})}"></span> על מנת לייצג את המאפיין העיקרי של קצב הגידול של הביטוי שבאגף שמאל. </p> <div class="mw-heading mw-heading3"><h3 id="חסם_על_קירובים"><span id=".D7.97.D7.A1.D7.9D_.D7.A2.D7.9C_.D7.A7.D7.99.D7.A8.D7.95.D7.91.D7.99.D7.9D"></span>חסם על קירובים</h3></div> <p>ידוע כי ניתן לתאר את פונקציית ה<a href="/wiki/%D7%90%D7%A7%D7%A1%D7%A4%D7%95%D7%A0%D7%A0%D7%98" title="אקספוננט">אקספוננט</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ e^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ e^{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92048c7dcca1097527e9189031b4cc9402115043" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.837ex; height:2.343ex;" alt="{\displaystyle \ e^{x}}"></span> באמצעות <a href="/wiki/%D7%98%D7%95%D7%A8_%D7%98%D7%99%D7%99%D7%9C%D7%95%D7%A8" title="טור טיילור">טור טיילור</a> אינסופי באופן הבא: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ e^{x}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>&#x2026;<!-- … --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ e^{x}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c0b9479152fcc64ac95cce4e878cb91ea592118" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:35.013ex; height:5.843ex;" alt="{\displaystyle \ e^{x}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\dots }"></span></li></ul> <p>כאשר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>x</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x\to 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/563049984fb0709025a3a547660e90612ff28ba7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.687ex; height:2.176ex;" alt="{\displaystyle \ x\to 0}"></span> ונרצה לקרב עד רמת דיוק של פולינום טיילור מסדר שני, דרך פשוטה לתאר ביטוי זה באמצעות חסם אסימפטוטי היא זו: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ e^{x}=1+x+{\frac {x^{2}}{2}}+o(x^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>o</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ e^{x}=1+x+{\frac {x^{2}}{2}}+o(x^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fc0b36564ed052b7aa14f15ccd0e2e66dc01dd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.489ex; height:5.676ex;" alt="{\displaystyle \ e^{x}=1+x+{\frac {x^{2}}{2}}+o(x^{2})}"></span></li></ul> <p>פירושו של סימון זה הוא כי ניתן לתאר את <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ e^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ e^{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92048c7dcca1097527e9189031b4cc9402115043" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.837ex; height:2.343ex;" alt="{\displaystyle \ e^{x}}"></span> באמצעות הפונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ 1+x+{\frac {x^{2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 1+x+{\frac {x^{2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db199df62f966fed9f7eee2a0bce365381bc5e6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.974ex; height:5.676ex;" alt="{\displaystyle \ 1+x+{\frac {x^{2}}{2}}}"></span> ועוד פונקציית "<a href="/wiki/%D7%A9%D7%90%D7%A8%D7%99%D7%AA_%D7%A9%D7%9C_%D7%98%D7%95%D7%A8_%D7%98%D7%99%D7%99%D7%9C%D7%95%D7%A8" title="שארית של טור טיילור">שארית</a>" ששייכת למחלקה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ o(x^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>o</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ o(x^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2536bc5b657ae8b4a5b41660ceca984db69110ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.901ex; height:3.176ex;" alt="{\displaystyle \ o(x^{2})}"></span>, כלומר היא זניחה יחסית ל-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ec7de2c20c0e309398fb4337b3f925cb618a36a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.965ex; height:2.676ex;" alt="{\displaystyle \ x^{2}}"></span> כאשר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>x</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x\to 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/563049984fb0709025a3a547660e90612ff28ba7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.687ex; height:2.176ex;" alt="{\displaystyle \ x\to 0}"></span> - בנוסחה: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to 0}{\frac {o(x^{2})}{x^{2}}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>o</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to 0}{\frac {o(x^{2})}{x^{2}}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d49ea8dc49ee8cde477f81b1f19c9f2afe3671eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.21ex; height:6.176ex;" alt="{\displaystyle \lim _{x\to 0}{\frac {o(x^{2})}{x^{2}}}=0}"></span> - כלומר, השארית זניחה ביחס לאיבר האחרון בטור שמחושב באופן מדויק. </p><p>במקרה זה נכונה גם טענה חזקה יותר: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ e^{x}=1+x+{\frac {x^{2}}{2}}+O(x^{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ e^{x}=1+x+{\frac {x^{2}}{2}}+O(x^{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41fe65ab58b2f5f06d3342d37a2edaf48062a0c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.135ex; height:5.676ex;" alt="{\displaystyle \ e^{x}=1+x+{\frac {x^{2}}{2}}+O(x^{3})}"></span></li></ul> <p>כאשר כאן השתמשנו ב-O גדול. </p> <div class="mw-heading mw-heading2"><h2 id="לקריאה_נוספת"><span id=".D7.9C.D7.A7.D7.A8.D7.99.D7.90.D7.94_.D7.A0.D7.95.D7.A1.D7.A4.D7.AA"></span>לקריאה נוספת</h2></div> <ul><li>קורמן, לייזרסון, ריבסט, <b>מבוא לאלגוריתמים</b>, <a href="/w/index.php?title=%D7%94%D7%95%D7%A6%D7%90%D7%AA_MIT&amp;action=edit&amp;redlink=1" class="new" title="הוצאת MIT (הדף אינו קיים)">הוצאת MIT</a>, תורגם לעברית על ידי <a href="/wiki/%D7%94%D7%90%D7%95%D7%A0%D7%99%D7%91%D7%A8%D7%A1%D7%99%D7%98%D7%94_%D7%94%D7%A4%D7%AA%D7%95%D7%97%D7%94" title="האוניברסיטה הפתוחה">האוניברסיטה הפתוחה</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="קישורים_חיצוניים"><span id=".D7.A7.D7.99.D7.A9.D7.95.D7.A8.D7.99.D7.9D_.D7.97.D7.99.D7.A6.D7.95.D7.A0.D7.99.D7.99.D7.9D"></span>קישורים חיצוניים</h2></div> <ul><li>גדי אלכסנדרוביץ', <a rel="nofollow" class="external text" href="https://gadial.net/2012/07/06/asymptotic_notation">הסבר בזמן (O(n על סימונים אסימפטוטיים</a>, באתר "לא מדויק", 6 ביולי 2012</li> <li><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Big-ONotation.html">סימון אסימפטוטי</a>, באתר <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a> <span dir="rtl" class="languageicon">(באנגלית)</span><style data-mw-deduplicate="TemplateStyles:r36549940">.mw-parser-output .languageicon{font-size:0.95em;font-weight:bold;color:#555}</style></li></ul> <p><br /> </p> <table class="navbox nowraplinks" style="width: 90%; clear: both; margin: 0.5em auto; margin-top: 0.5em; margin-bottom: 0.5em; padding: 0.2em; text-align: right;"> <tbody><tr> <th colspan="3" style="text-align: center; padding-top: 0.1em; padding-bottom: 0.1em; color: black; background:#d1eeee; font-weight: bold;"><a href="/wiki/%D7%90%D7%9C%D7%92%D7%95%D7%A8%D7%99%D7%AA%D7%9D_%D7%9E%D7%99%D7%95%D7%9F" title="אלגוריתם מיון">אלגוריתמי מיון</a> </th></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">רקע תאורטי </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%A1%D7%99%D7%91%D7%95%D7%9B%D7%99%D7%95%D7%AA" title="תורת הסיבוכיות">תורת הסיבוכיות</a> • <a class="mw-selflink selflink">סימון אסימפטוטי</a> • <a href="/wiki/%D7%A1%D7%93%D7%A8_%D7%9E%D7%9C%D7%90" title="סדר מלא">סדר מלא</a> • <a href="/wiki/%D7%A8%D7%A9%D7%99%D7%9E%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%A0%D7%AA%D7%95%D7%A0%D7%99%D7%9D)" title="רשימה (מבנה נתונים)">רשימה</a> • <a href="/wiki/%D7%90%D7%9C%D7%92%D7%95%D7%A8%D7%99%D7%AA%D7%9D_%D7%AA%D7%95%D7%9A-%D7%9E%D7%A7%D7%95%D7%9E%D7%99" title="אלגוריתם תוך-מקומי">אלגוריתם תוך-מקומי</a> • <a href="/wiki/%D7%9E%D7%99%D7%95%D7%9F_%D7%99%D7%A6%D7%99%D7%91" title="מיון יציב">מיון יציב</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;"><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%90%D7%9C%D7%92%D7%95%D7%A8%D7%99%D7%AA%D7%9E%D7%99_%D7%9E%D7%99%D7%95%D7%9F" title="קטגוריה:אלגוריתמי מיון">אלגוריתמי מיון</a> </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%9F_%D7%90%D7%A7%D7%A8%D7%90%D7%99" title="מיון אקראי">אקראי</a> • <a href="/wiki/%D7%9E%D7%99%D7%95%D7%9F_%D7%91%D7%95%D7%A2%D7%95%D7%AA" title="מיון בועות">בועות</a> • <a href="/wiki/%D7%9E%D7%99%D7%95%D7%9F_%D7%91%D7%97%D7%99%D7%A8%D7%94" title="מיון בחירה">בחירה</a> • <a href="/wiki/%D7%9E%D7%99%D7%95%D7%9F_%D7%91%D7%A1%D7%99%D7%A1" title="מיון בסיס">בסיס</a> • <a href="/wiki/%D7%9E%D7%99%D7%95%D7%9F_%D7%94%D7%9B%D7%A0%D7%A1%D7%94" title="מיון הכנסה">הכנסה</a> • <a href="/wiki/%D7%9E%D7%99%D7%95%D7%9F_%D7%9E%D7%94%D7%99%D7%A8" title="מיון מהיר">מהיר</a> • <a href="/wiki/%D7%9E%D7%99%D7%95%D7%9F_%D7%9E%D7%99%D7%96%D7%95%D7%92" title="מיון מיזוג">מיזוג</a> • <a href="/wiki/%D7%9E%D7%99%D7%95%D7%9F_%D7%9E%D7%A0%D7%99%D7%99%D7%94" title="מיון מנייה">מנייה</a> • <a href="/wiki/%D7%9E%D7%99%D7%95%D7%9F_%D7%9E%D7%A1%D7%A8%D7%A7" title="מיון מסרק">מסרק</a> • <a href="/wiki/%D7%9E%D7%99%D7%95%D7%9F_%D7%A1%D7%9C%D7%99%D7%9D" title="מיון סלים">סלים</a> • <a href="/wiki/%D7%9E%D7%99%D7%95%D7%9F_%D7%A2%D7%A8%D7%99%D7%9E%D7%94" title="מיון ערימה">ערימה</a> • <a href="/wiki/%D7%A8%D7%A9%D7%AA_%D7%9E%D7%99%D7%95%D7%9F" title="רשת מיון">רשת מיון</a> • <a href="/wiki/%D7%9E%D7%99%D7%95%D7%9F_%D7%A9%D7%99%D7%99%D7%A7%D7%A8" title="מיון שייקר">שייקר</a> • <a href="/wiki/%D7%9E%D7%99%D7%95%D7%9F_%D7%A9%D7%9C" title="מיון של">של</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">שונות </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%9F_%D7%98%D7%95%D7%A4%D7%95%D7%9C%D7%95%D7%92%D7%99" title="מיון טופולוגי">מיון טופולוגי</a> • <a href="/wiki/%D7%91%D7%A2%D7%99%D7%99%D7%AA_%D7%A1%D7%99%D7%93%D7%95%D7%A8_%D7%94%D7%A4%D7%A0%D7%A7%D7%99%D7%99%D7%A7%D7%99%D7%9D" title="בעיית סידור הפנקייקים">בעיית סידור הפנקייקים של גודמן</a> </td></tr> </tbody></table> <!-- NewPP limit report Parsed by mw‐web.eqiad.main‐68d6679558‐7vjds Cached time: 20241106202601 Cache expiry: 2592000 Reduced expiry: false Complications: [show‐toc] CPU time usage: 0.153 seconds Real time usage: 0.317 seconds Preprocessor visited node count: 1149/1000000 Post‐expand include size: 6649/2097152 bytes Template argument size: 3345/2097152 bytes Highest expansion depth: 10/100 Expensive parser function count: 8/500 Unstrip recursion depth: 0/20 Unstrip post‐expand size: 4357/5000000 bytes Lua time usage: 0.019/10.000 seconds Lua memory usage: 1172353/52428800 bytes Number of Wikibase entities loaded: 1/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 101.285 1 -total 57.41% 58.146 1 תבנית:MathWorld 29.79% 30.176 1 תבנית:מילה_ראשונה 26.53% 26.872 2 תבנית:ויקינתונים 17.81% 18.039 1 תבנית:עוגן 12.95% 13.115 1 תבנית:אלגוריתמי_מיון 11.35% 11.494 1 תבנית:אנגלית 10.65% 10.787 1 תבנית:ניווט_קבוצות 9.89% 10.015 1 תבנית:שפת_קישור 9.88% 10.003 1 תבנית:לא_מדויק --> <!-- Saved in parser cache with key hewiki:pcache:idhash:274565-0!canonical and timestamp 20241106202601 and revision id 39136980. Rendering was triggered because: page-view --> </div><!--esi <esi:include src="/esitest-fa8a495983347898/content" /> --><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">אוחזר מתוך "<a dir="ltr" href="https://he.wikipedia.org/w/index.php?title=סימון_אסימפטוטי&amp;oldid=39136980">https://he.wikipedia.org/w/index.php?title=סימון_אסימפטוטי&amp;oldid=39136980</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94" title="ויקיפדיה:קטגוריה">קטגוריות</a>: <ul><li><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%A1%D7%99%D7%9E%D7%A0%D7%99%D7%9D_%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%99%D7%9D" title="קטגוריה:סימנים מתמטיים">סימנים מתמטיים</a></li><li><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%90%D7%9C%D7%92%D7%95%D7%A8%D7%99%D7%AA%D7%9E%D7%99%D7%9D" title="קטגוריה:אלגוריתמים">אלגוריתמים</a></li><li><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%A1%D7%99%D7%91%D7%95%D7%9B%D7%99%D7%95%D7%AA_%D7%97%D7%99%D7%A9%D7%95%D7%91%D7%99%D7%AA" title="קטגוריה:סיבוכיות חישובית">סיבוכיות חישובית</a></li></ul></div></div> </div> </main> <div id='mw-data-after-content'> <div class="read-more-container"></div> </div> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> דף זה נערך לאחרונה ב־29 ביולי 2024, בשעה 22:51.</li> <li id="footer-info-copyright">הטקסט מוגש בכפוף לרישיון <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.he">Creative Commons ייחוס-שיתוף זהה 4.0</a>; ייתכן שישנם תנאים נוספים. ר׳ את <a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use">תנאי השימוש</a> לפרטים.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">מדיניות פרטיות</a></li> <li id="footer-places-about"><a href="/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:%D7%90%D7%95%D7%93%D7%95%D7%AA">אודות ויקיפדיה</a></li> <li id="footer-places-disclaimers"><a href="/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:%D7%94%D7%91%D7%94%D7%A8%D7%95%D7%AA_%D7%9E%D7%A9%D7%A4%D7%98%D7%99%D7%95%D7%AA">הבהרות משפטיות</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">קוד התנהגות</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">מפתחים</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/he.wikipedia.org">סטטיסטיקות</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">הצהרה על עוגיות</a></li> <li id="footer-places-mobileview"><a href="//he.m.wikipedia.org/w/index.php?title=%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99&amp;printable=yes&amp;mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">תצוגת מכשירים ניידים</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-7dfb9d98f5-dvj4t","wgBackendResponseTime":160,"wgPageParseReport":{"limitreport":{"cputime":"0.153","walltime":"0.317","ppvisitednodes":{"value":1149,"limit":1000000},"postexpandincludesize":{"value":6649,"limit":2097152},"templateargumentsize":{"value":3345,"limit":2097152},"expansiondepth":{"value":10,"limit":100},"expensivefunctioncount":{"value":8,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":4357,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 101.285 1 -total"," 57.41% 58.146 1 תבנית:MathWorld"," 29.79% 30.176 1 תבנית:מילה_ראשונה"," 26.53% 26.872 2 תבנית:ויקינתונים"," 17.81% 18.039 1 תבנית:עוגן"," 12.95% 13.115 1 תבנית:אלגוריתמי_מיון"," 11.35% 11.494 1 תבנית:אנגלית"," 10.65% 10.787 1 תבנית:ניווט_קבוצות"," 9.89% 10.015 1 תבנית:שפת_קישור"," 9.88% 10.003 1 תבנית:לא_מדויק"]},"scribunto":{"limitreport-timeusage":{"value":"0.019","limit":"10.000"},"limitreport-memusage":{"value":1172353,"limit":52428800}},"cachereport":{"origin":"mw-web.eqiad.main-68d6679558-7vjds","timestamp":"20241106202601","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"\u05e1\u05d9\u05de\u05d5\u05df \u05d0\u05e1\u05d9\u05de\u05e4\u05d8\u05d5\u05d8\u05d9","url":"https:\/\/he.wikipedia.org\/wiki\/%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%90%D7%A1%D7%99%D7%9E%D7%A4%D7%98%D7%95%D7%98%D7%99","sameAs":"http:\/\/www.wikidata.org\/entity\/Q269878","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q269878","author":{"@type":"Organization","name":"\u05ea\u05d5\u05e8\u05de\u05d9\u05dd \u05dc\u05de\u05d9\u05d6\u05de\u05d9 \u05d5\u05d9\u05e7\u05d9\u05de\u05d3\u05d9\u05d4"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2006-11-24T11:06:08Z","dateModified":"2024-07-29T20:51:33Z"}</script> </body> </html>

Pages： 1 2 3 4 5 6 7 8 9 10