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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&utm_medium=sidebar&utm_campaign=C13_he.wikipedia.org&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&returnto=%D7%92%D7%91%D7%95%D7%9C+%D7%A9%D7%9C+%D7%A1%D7%93%D7%A8%D7%94" 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&returnto=%D7%92%D7%91%D7%95%D7%9C+%D7%A9%D7%9C+%D7%A1%D7%93%D7%A8%D7%94" 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 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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-סימון" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#סימון"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>סימון</span> </div> </a> <ul id="toc-סימון-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-הגדרת_הגבול" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#הגדרת_הגבול"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>הגדרת הגבול</span> </div> </a> <button aria-controls="toc-הגדרת_הגבול-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 הגדרת הגבול</span> </button> <ul id="toc-הגדרת_הגבול-sublist" class="vector-toc-list"> <li id="toc-המשמעות_של_"קירבה_לגבול"" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#המשמעות_של_"קירבה_לגבול""> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>המשמעות של "קירבה לגבול"</span> </div> </a> <ul id="toc-המשמעות_של_"קירבה_לגבול"-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-המשמעות_של_"התקרבות_לגבול"" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#המשמעות_של_"התקרבות_לגבול""> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>המשמעות של "התקרבות לגבול"</span> </div> </a> <ul id="toc-המשמעות_של_"התקרבות_לגבול"-sublist" class="vector-toc-list"> <li id="toc-איתור_המספר_'"`UNIQ--postMath-00000034-QINU`"'_שממנו_והלאה_אברי_הסדרה_נמצאים_בסביבת_הגבול" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#איתור_המספר_'"`UNIQ--postMath-00000034-QINU`"'_שממנו_והלאה_אברי_הסדרה_נמצאים_בסביבת_הגבול"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>איתור המספר '"`UNIQ--postMath-00000034-QINU`"' שממנו והלאה אברי הסדרה נמצאים בסביבת הגבול</span> </div> </a> <ul id="toc-איתור_המספר_'"`UNIQ--postMath-00000034-QINU`"'_שממנו_והלאה_אברי_הסדרה_נמצאים_בסביבת_הגבול-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-אפיון_התכנסות_לפי_קושי" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#אפיון_התכנסות_לפי_קושי"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>אפיון התכנסות לפי קושי</span> </div> </a> <ul id="toc-אפיון_התכנסות_לפי_קושי-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-גבול_במובן_הרחב" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#גבול_במובן_הרחב"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>גבול במובן הרחב</span> </div> </a> <ul id="toc-גבול_במובן_הרחב-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-הגבול_כאופרטור" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#הגבול_כאופרטור"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>הגבול כאופרטור</span> </div> </a> <ul id="toc-הגבול_כאופרטור-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-מושגים_קרובים" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#מושגים_קרובים"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>מושגים קרובים</span> </div> </a> <ul id="toc-מושגים_קרובים-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-ראו_גם" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#ראו_גם"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>ראו גם</span> </div> </a> <ul id="toc-ראו_גם-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">6</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">גבול של סדרה</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־42 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-42" 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">42 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/Limit_of_a_sequence" title="Limit of a sequence – אנגלית" lang="en" hreflang="en" data-title="Limit of a sequence" 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/%D9%86%D9%87%D8%A7%D9%8A%D8%A9_%D9%85%D8%AA%D8%AA%D8%A7%D9%84%D9%8A%D8%A9" title="نهاية متتالية – ערבית" lang="ar" hreflang="ar" data-title="نهاية متتالية" data-language-autonym="العربية" data-language-local-name="ערבית" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Llende_d%27una_socesi%C3%B3n" title="Llende d'una socesión – אסטורית" lang="ast" hreflang="ast" data-title="Llende d'una socesión" data-language-autonym="Asturianu" data-language-local-name="אסטורית" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%AD%D2%99%D0%BC%D3%99-%D1%8D%D2%99%D0%BB%D0%B5%D0%BB%D0%B5%D0%BA%D1%82%D0%B5%D2%A3_%D1%81%D0%B8%D0%BA%D0%BB%D3%99%D0%BD%D0%BC%D3%99%D2%BB%D0%B5" title="Эҙмә-эҙлелектең сикләнмәһе – בשקירית" lang="ba" hreflang="ba" data-title="Эҙмә-эҙлелектең сикләнмәһе" data-language-autonym="Башҡортса" data-language-local-name="בשקירית" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9B%D1%96%D0%BC%D1%96%D1%82_%D0%BF%D0%B0%D1%81%D0%BB%D1%8F%D0%B4%D0%BE%D1%9E%D0%BD%D0%B0%D1%81%D1%86%D1%96" title="Ліміт паслядоўнасці – בלארוסית" lang="be" hreflang="be" data-title="Ліміт паслядоўнасці" data-language-autonym="Беларуская" data-language-local-name="בלארוסית" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D0%B0_%D0%BD%D0%B0_%D1%80%D0%B5%D0%B4%D0%B8%D1%86%D0%B0" title="Граница на редица – בולגרית" lang="bg" hreflang="bg" data-title="Граница на редица" data-language-autonym="Български" data-language-local-name="בולגרית" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A7%8D%E0%A6%B0%E0%A6%AE_%E0%A6%B8%E0%A7%80%E0%A6%AE%E0%A6%BE" title="ক্রম সীমা – בנגלית" lang="bn" hreflang="bn" data-title="ক্রম সীমা" data-language-autonym="বাংলা" data-language-local-name="בנגלית" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Grani%C4%8Dna_vrijednost_niza" title="Granična vrijednost niza – בוסנית" lang="bs" hreflang="bs" data-title="Granična vrijednost niza" data-language-autonym="Bosanski" data-language-local-name="בוסנית" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Converg%C3%A8ncia_(successi%C3%B3_matem%C3%A0tica)" title="Convergència (successió matemàtica) – קטלאנית" lang="ca" hreflang="ca" data-title="Convergència (successió matemàtica)" 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/Limita_posloupnosti" title="Limita posloupnosti – צ׳כית" lang="cs" hreflang="cs" data-title="Limita posloupnosti" data-language-autonym="Čeština" data-language-local-name="צ׳כית" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A3%D0%BC%D0%BB%C4%83%D0%BD-%D1%85%D1%8B%C3%A7%D0%BB%C4%83%D0%BD%D0%BB%C4%83%D1%85_%D1%87%D0%B8%D0%BA%D0%BA%D0%B8" title="Умлăн-хыçлăнлăх чикки – צ׳ובשית" lang="cv" hreflang="cv" data-title="Умлăн-хыçлăнлăх чикки" data-language-autonym="Чӑвашла" data-language-local-name="צ׳ובשית" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Grenzwert_(Folge)" title="Grenzwert (Folge) – גרמנית" lang="de" hreflang="de" data-title="Grenzwert (Folge)" data-language-autonym="Deutsch" data-language-local-name="גרמנית" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%8C%CF%81%CE%B9%CE%BF_%CE%B1%CE%BA%CE%BF%CE%BB%CE%BF%CF%85%CE%B8%CE%AF%CE%B1%CF%82" title="Όριο ακολουθίας – יוונית" lang="el" hreflang="el" data-title="Όριο ακολουθίας" data-language-autonym="Ελληνικά" data-language-local-name="יוונית" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/L%C3%ADmite_de_una_sucesi%C3%B3n" title="Límite de una sucesión – ספרדית" lang="es" hreflang="es" data-title="Límite de una sucesión" data-language-autonym="Español" data-language-local-name="ספרדית" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Jada_piirv%C3%A4%C3%A4rtus" title="Jada piirväärtus – אסטונית" lang="et" hreflang="et" data-title="Jada piirväärtus" data-language-autonym="Eesti" data-language-local-name="אסטונית" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Segida_baten_limite" title="Segida baten limite – בסקית" lang="eu" hreflang="eu" data-title="Segida baten limite" 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/%D8%AD%D8%AF_%D8%AF%D9%86%D8%A8%D8%A7%D9%84%D9%87" title="حد دنباله – פרסית" lang="fa" hreflang="fa" data-title="حد دنباله" data-language-autonym="فارسی" data-language-local-name="פרסית" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Lukujonon_raja-arvo" title="Lukujonon raja-arvo – פינית" lang="fi" hreflang="fi" data-title="Lukujonon raja-arvo" data-language-autonym="Suomi" data-language-local-name="פינית" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Limite_d%27une_suite" title="Limite d'une suite – צרפתית" lang="fr" hreflang="fr" data-title="Limite d'une suite" data-language-autonym="Français" data-language-local-name="צרפתית" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/L%C3%ADmite_dunha_sucesi%C3%B3n" title="Límite dunha sucesión – גליסית" lang="gl" hreflang="gl" data-title="Límite dunha sucesión" data-language-autonym="Galego" data-language-local-name="גליסית" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%A8%E0%A5%81%E0%A4%95%E0%A5%8D%E0%A4%B0%E0%A4%AE_%E0%A4%95%E0%A5%80_%E0%A4%B8%E0%A5%80%E0%A4%AE%E0%A4%BE" title="अनुक्रम की सीमा – הינדי" lang="hi" hreflang="hi" data-title="अनुक्रम की सीमा" data-language-autonym="हिन्दी" data-language-local-name="הינדי" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Limit_barisan" title="Limit barisan – אינדונזית" lang="id" hreflang="id" data-title="Limit barisan" 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/Limite_di_una_successione" title="Limite di una successione – איטלקית" lang="it" hreflang="it" data-title="Limite di una successione" 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/%E6%95%B0%E5%88%97%E3%81%AE%E6%A5%B5%E9%99%90" title="数列の極限 – יפנית" lang="ja" hreflang="ja" data-title="数列の極限" 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-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%88%98%EC%97%B4%EC%9D%98_%EA%B7%B9%ED%95%9C" title="수열의 극한 – קוריאנית" lang="ko" hreflang="ko" data-title="수열의 극한" data-language-autonym="한국어" data-language-local-name="קוריאנית" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Limit_di_succession" title="Limit di succession – לומברדית" lang="lmo" hreflang="lmo" data-title="Limit di succession" data-language-autonym="Lombard" data-language-local-name="לומברדית" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Sekos_riba" title="Sekos riba – ליטאית" lang="lt" hreflang="lt" data-title="Sekos riba" data-language-autonym="Lietuvių" data-language-local-name="ליטאית" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%BD%D0%B8%D1%87%D0%BD%D0%B0_%D0%B2%D1%80%D0%B5%D0%B4%D0%BD%D0%BE%D1%81%D1%82_%D0%BD%D0%B0_%D0%BD%D0%B8%D0%B7%D0%B0" title="Гранична вредност на низа – מקדונית" lang="mk" hreflang="mk" data-title="Гранична вредност на низа" data-language-autonym="Македонски" data-language-local-name="מקדונית" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Granica_ci%C4%85gu" title="Granica ciągu – פולנית" lang="pl" hreflang="pl" data-title="Granica ciągu" 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/Limite_de_uma_sequ%C3%AAncia" title="Limite de uma sequência – פורטוגזית" lang="pt" hreflang="pt" data-title="Limite de uma sequência" 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/Limit%C4%83_a_unui_%C8%99ir" title="Limită a unui șir – רומנית" lang="ro" hreflang="ro" data-title="Limită a unui șir" 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/%D0%9F%D1%80%D0%B5%D0%B4%D0%B5%D0%BB_%D0%BF%D0%BE%D1%81%D0%BB%D0%B5%D0%B4%D0%BE%D0%B2%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D0%B8" title="Предел последовательности – רוסית" lang="ru" hreflang="ru" data-title="Предел последовательности" 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/Limit_of_a_sequence" title="Limit of a sequence – אנגלית פשוטה" lang="en-simple" hreflang="en-simple" data-title="Limit of a sequence" 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/Limita_zaporedja" title="Limita zaporedja – סלובנית" lang="sl" hreflang="sl" data-title="Limita zaporedja" 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%93%D1%80%D0%B0%D0%BD%D0%B8%D1%87%D0%BD%D0%B0_%D0%B2%D1%80%D0%B5%D0%B4%D0%BD%D0%BE%D1%81%D1%82_%D0%BD%D0%B8%D0%B7%D0%B0" title="Гранична вредност низа – סרבית" lang="sr" hreflang="sr" data-title="Гранична вредност низа" 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-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AF%8A%E0%AE%9F%E0%AE%B0%E0%AF%8D%E0%AE%B5%E0%AE%B0%E0%AE%BF%E0%AE%9A%E0%AF%88%E0%AE%AF%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%8E%E0%AE%B2%E0%AF%8D%E0%AE%B2%E0%AF%88" title="தொடர்வரிசையின் எல்லை – טמילית" lang="ta" hreflang="ta" data-title="தொடர்வரிசையின் எல்லை" data-language-autonym="தமிழ்" data-language-local-name="טמילית" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A5%E0%B8%B4%E0%B8%A1%E0%B8%B4%E0%B8%95%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%A5%E0%B8%B3%E0%B8%94%E0%B8%B1%E0%B8%9A" title="ลิมิตของลำดับ – תאית" lang="th" hreflang="th" data-title="ลิมิตของลำดับ" 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/Dizinin_limiti" title="Dizinin limiti – טורקית" lang="tr" hreflang="tr" data-title="Dizinin limiti" 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%93%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D1%8F_%D0%BF%D0%BE%D1%81%D0%BB%D1%96%D0%B4%D0%BE%D0%B2%D0%BD%D0%BE%D1%81%D1%82%D1%96" title="Границя послідовності – אוקראינית" lang="uk" hreflang="uk" data-title="Границя послідовності" 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/Gi%E1%BB%9Bi_h%E1%BA%A1n_c%E1%BB%A7a_m%E1%BB%99t_d%C3%A3y" title="Giới hạn của một dãy – וייטנאמית" lang="vi" hreflang="vi" data-title="Giới hạn của một dãy" 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/%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90" title="数列极限 – סינית" lang="zh" hreflang="zh" data-title="数列极限" 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/%E6%A5%B5%E9%99%90_(%E6%95%B8%E5%88%97)" title="極限 (數列) – קנטונזית" lang="yue" hreflang="yue" data-title="極限 (數列)" 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/Q847204#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%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94" 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%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94" 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%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94"><span>קריאה</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&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%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&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%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&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%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94"><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%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&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%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&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%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&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%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94" 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%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94" 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%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&oldid=37909554" 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%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&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&page=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&id=37909554&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&url=https%3A%2F%2Fhe.wikipedia.org%2Fwiki%2F%25D7%2592%25D7%2591%25D7%2595%25D7%259C_%25D7%25A9%25D7%259C_%25D7%25A1%25D7%2593%25D7%25A8%25D7%2594"><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&url=https%3A%2F%2Fhe.wikipedia.org%2Fwiki%2F%25D7%2592%25D7%2591%25D7%2595%25D7%259C_%25D7%25A9%25D7%259C_%25D7%25A1%25D7%2593%25D7%25A8%25D7%2594"><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&bookcmd=book_creator&referer=%D7%92%D7%91%D7%95%D7%9C+%D7%A9%D7%9C+%D7%A1%D7%93%D7%A8%D7%94"><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&page=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&action=show-download-screen"><span>הורדה כ־PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&printable=yes" title="גרסה להדפסה של הדף הזה [p]" accesskey="p"><span>גרסה להדפסה</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 class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Limit_of_a_sequence" hreflang="en"><span>ויקישיתוף</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q847204" 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="mw-content-rtl mw-parser-output" lang="he" dir="rtl"><div class="noexcerpt noprint dablink" style="font-size: 90%; color: #555577;margin-right:22px;"><span typeof="mw:File"><span title="פירוש נוסף"><img alt="פירוש נוסף" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Disambig_RTL.svg/25px-Disambig_RTL.svg.png" decoding="async" width="25" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Disambig_RTL.svg/38px-Disambig_RTL.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Disambig_RTL.svg/50px-Disambig_RTL.svg.png 2x" data-file-width="220" data-file-height="168" /></span></span> ערך זה עוסק באיבר הגבול של סדרה מתכנסת. אם התכוונתם לסדרה שיש לה גבול, ראו <span class="nodisambig"><a href="/wiki/%D7%A1%D7%93%D7%A8%D7%94_%D7%9E%D7%AA%D7%9B%D7%A0%D7%A1%D7%AA" title="סדרה מתכנסת">סדרה מתכנסת</a></span>.</div> <table cellpadding="1" style="float: left; clear:both; border: 1px solid #8888aa; background: #f7f8ff; padding: 5px; font-size: 95%; margin: 0.5em 1em 0.5em 0.5em;"> <tbody><tr> <td style="text-align: center;"><span typeof="mw:File"><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"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/60px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/90px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/120px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> <p>בערך זה<br />נעשה שימוש<br />בסימנים מוסכמים<br />מתחום המתמטיקה.<br />להבהרת הסימנים<br />ראו <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 מתמטי</a>. </p> </td></tr></tbody></table> <p>ב<a href="/wiki/%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%90%D7%99%D7%A0%D7%A4%D7%99%D7%A0%D7%99%D7%98%D7%A1%D7%99%D7%9E%D7%9C%D7%99" title="חשבון אינפיניטסימלי">חשבון אינפיניטסימלי</a>, <b>גבול</b> של <a href="/wiki/%D7%A1%D7%93%D7%A8%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="סדרה (מתמטיקה)">סדרה</a> <a href="/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%9E%D7%9E%D7%A9%D7%99" title="מספר ממשי">ממשית</a> הוא מספר, ש<a href="/wiki/%D7%90%D7%99%D7%91%D7%A8_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" class="mw-redirect" title="איבר (מתמטיקה)">איברי</a> הסדרה הולכים ומתקרבים אליו כך שה<a href="/wiki/%D7%9E%D7%A8%D7%97%D7%A7" title="מרחק">מרחק</a> בין האיברים לגבול קטן כרצוננו. מושג זה מהווה אבן פינה ב<a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%AA" title="אנליזה מתמטית">אנליזה המתמטית</a>, בכך שהוא מאפשר לנסח ולחקור בכלים סופיים את ההתנהגות של סדרות, <a href="/wiki/%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94" title="גבול של פונקציה">פונקציות</a> ותהליכים אינסופיים אחרים. סדרה שיש לה גבול נקראת <b><a href="/wiki/%D7%A1%D7%93%D7%A8%D7%94_%D7%9E%D7%AA%D7%9B%D7%A0%D7%A1%D7%AA" title="סדרה מתכנסת">סדרה מתכנסת</a></b>. סדרה שאין לה גבול (אינה מתכנסת) נקראת <b>סדרה מתבדרת</b>. </p><p>באופן כללי יותר, אפשר להגדיר גבול לסדרה גם אם אבריה אינם דווקא מספרים ממשיים: ראו <a href="/wiki/%D7%92%D7%91%D7%95%D7%9C_(%D7%98%D7%95%D7%A4%D7%95%D7%9C%D7%95%D7%92%D7%99%D7%94)" title="גבול (טופולוגיה)">גבול בטופולוגיה</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="סימון"><span id=".D7.A1.D7.99.D7.9E.D7.95.D7.9F"></span>סימון</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&action=edit&section=1" title="עריכת קוד המקור של הפרק: סימון"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&veaction=edit&section=1" title="עריכת פסקה: "סימון"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>את הטענה "הגבול של הסדרה <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 \{a_{n}\}_{n=1}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a_{n}\}_{n=1}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a28a0eda962ff3986e2b8abba0cad913ee837c75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.092ex; height:3.009ex;" alt="{\displaystyle \{a_{n}\}_{n=1}^{\infty }}"></span> הוא <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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span>" ניתן לכתוב באופן <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 }a_{n}=L}"> <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">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }a_{n}=L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec8a6da951ff8db5ca56f8e9454ed61271cf6ffd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.789ex; height:3.676ex;" alt="{\displaystyle \lim _{n\to \infty }a_{n}=L}"></span> או <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 a_{n}\to L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}\to L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11b96c0ee27d23b06518212eb970a21743f13383" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.645ex; height:2.509ex;" alt="{\displaystyle a_{n}\to L}"></span> (ולפעמים גם <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 a_{n}\xrightarrow {n\to \infty } L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mpadded> </mover> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}\xrightarrow {n\to \infty } L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/638e33d183edc0b5a02185127d609c2c2f920102" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.445ex; width:10.918ex; height:3.843ex;" alt="{\displaystyle a_{n}\xrightarrow {n\to \infty } L}"></span>). הצירוף lim הוא קיצור של המילה הלטינית limes שפירושה "גבול". 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 a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span>, האות <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> מציינת את הסדרה שהאיבר שייך אליה, 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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> הוא <a href="/wiki/%D7%90%D7%99%D7%A0%D7%93%D7%A7%D7%A1_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="אינדקס (מתמטיקה)">אינדקס</a>, המציין את מספרו הסידורי של האיבר בסדרה. הסמל <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 \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }"></span> מסמל את מושג ה<a href="/wiki/%D7%90%D7%99%D7%A0%D7%A1%D7%95%D7%A3" title="אינסוף">אינסוף</a>. החץ <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 \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"></span> מסמל שאיפה של הביטוי המצוין בתחילת החץ לזה שבסופו. כך 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\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d55d9b32f6fa8fab6a84ea444a6b5a24bb45e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\to \infty }"></span> מציין 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"> <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/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> שואף לאינסוף. </p> <div class="mw-heading mw-heading2"><h2 id="הגדרת_הגבול"><span id=".D7.94.D7.92.D7.93.D7.A8.D7.AA_.D7.94.D7.92.D7.91.D7.95.D7.9C"></span>הגדרת הגבול</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&action=edit&section=2" title="עריכת קוד המקור של הפרק: הגדרת הגבול"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&veaction=edit&section=2" title="עריכת פסקה: "הגדרת הגבול"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>אינטואיטיבית, מספר ממשי הוא גבול של סדרת מספרים אם אברי הסדרה הולכים ומתקרבים אליו. עם זאת הגדרה זו מעורפלת ואינה שימושית. כדי לתת הגדרה מדויקת יותר יש להשתמש במושגים של סביבה ומרחק, שיחולו על אברי הסדרה ממקום מסוים ואילך. </p> <div class="mw-heading mw-heading3"><h3 id="המשמעות_של_"קירבה_לגבול""><span id=".D7.94.D7.9E.D7.A9.D7.9E.D7.A2.D7.95.D7.AA_.D7.A9.D7.9C_.22.D7.A7.D7.99.D7.A8.D7.91.D7.94_.D7.9C.D7.92.D7.91.D7.95.D7.9C.22"></span>המשמעות של "קירבה לגבול"</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&action=edit&section=3" title="עריכת קוד המקור של הפרק: המשמעות של "קירבה לגבול""><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&veaction=edit&section=3" title="עריכת פסקה: "המשמעות של "קירבה לגבול""" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5:Epsilon_Umgebung.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Epsilon_Umgebung.svg/250px-Epsilon_Umgebung.svg.png" decoding="async" width="250" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Epsilon_Umgebung.svg/375px-Epsilon_Umgebung.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Epsilon_Umgebung.svg/500px-Epsilon_Umgebung.svg.png 2x" data-file-width="699" data-file-height="67" /></a><figcaption>ייצוג גרפי של סביבת הנקודה <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> על ישר המספרים. המרחק של כל מספר בסביבה מהנקודה <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> קטן מרדיוס הסביבה, המיוצג על ידי <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 \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></span> ויכול להיות קטן כרצוננו.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5:Geometrical_interpretation_of_the_absolute_value.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Geometrical_interpretation_of_the_absolute_value.svg/250px-Geometrical_interpretation_of_the_absolute_value.svg.png" decoding="async" width="250" height="69" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Geometrical_interpretation_of_the_absolute_value.svg/375px-Geometrical_interpretation_of_the_absolute_value.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Geometrical_interpretation_of_the_absolute_value.svg/500px-Geometrical_interpretation_of_the_absolute_value.svg.png 2x" data-file-width="360" data-file-height="100" /></a><figcaption>על גבי <a href="/wiki/%D7%94%D7%99%D7%A9%D7%A8_%D7%94%D7%9E%D7%9E%D7%A9%D7%99" title="הישר הממשי">הישר הממשי</a> המרחק בין שני מספרים מוגדר כ<a href="/wiki/%D7%A2%D7%A8%D7%9A_%D7%9E%D7%95%D7%97%D7%9C%D7%98" title="ערך מוחלט">ערך המוחלט</a> של ה<a href="/wiki/%D7%97%D7%99%D7%A1%D7%95%D7%A8" title="חיסור">הפרש</a> שלהם</figcaption></figure> <p>קבוצת המספרים שמרחקם מערך נתון <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> אינו עולה על גודל קבוע (כלשהו), נקראת <b><a href="/wiki/%D7%A1%D7%91%D7%99%D7%91%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="סביבה (מתמטיקה)">סביבה</a></b> 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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>. בהקשר זה, מקובל להשתמש באות היוונית אפסילון (<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 \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></span>) כדי לייצג את רדיוס הסביבה. רדיוס הסביבה הוא <a href="/wiki/%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D_%D7%97%D7%99%D7%95%D7%91%D7%99%D7%99%D7%9D_%D7%95%D7%A9%D7%9C%D7%99%D7%9C%D7%99%D7%99%D7%9D" title="מספרים חיוביים ושליליים">מספר חיובי</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 \varepsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04ec3670b50384a3ce48aca42e7cc5131a06b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}"></span>), אשר יכול להיות קטן כרצוננו. סביבה זו כוללת את כל המספרים <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> המקיימים את התנאי <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-a|<\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x-a|<\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e5c4161ae92084da0ba0509d774e29ab7dfee7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.876ex; height:2.843ex;" alt="{\displaystyle |x-a|<\varepsilon }"></span>. </p><p>במסגרת מושגים אלו ניתן להגדיר את אברי הסדרה ה"קרובים" אל הגבול בצורה מדויקת יותר כנמצאים בסביבה של הגבול, כך שהמרחק 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 a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span> 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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> יהיה קטן 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 \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></span>; כלומר <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 \left|a_{n}-L\right|<\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>L</mi> </mrow> <mo>|</mo> </mrow> <mo><</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|a_{n}-L\right|<\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/115cbf6ce4bb333b0b0b1d0b24f55ebd84d96d4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.347ex; height:2.843ex;" alt="{\displaystyle \left|a_{n}-L\right|<\varepsilon }"></span>. </p><p>משום 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 \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></span> יכול להיות קטן כרצוננו, על אברי הסדרה לענות על תנאי זה עבור כל <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 \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></span>. הסימון המתמטי של "לכל" הוא <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 \forall }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc1a1a9c4c0f8d5df989c98aa2773ed657c5937" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \forall }"></span>. על כן, נכתוב את התנאי "לכל רדיוס סביבה חיובי אברי הסדרה נמצאים בסביבת הגבול" בצורה הבאה: <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 \forall \varepsilon >0,\,|a_{n}-L|<\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>ε</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \varepsilon >0,\,|a_{n}-L|<\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc2342318cf3a851cfd7fe6b1c49b1404ef609d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.405ex; height:2.843ex;" alt="{\displaystyle \forall \varepsilon >0,\,|a_{n}-L|<\varepsilon }"></span>. </p><p>עם זאת, משום שהגדרנו את אברי הסדרה כ"הולכים ומתקרבים" לגבול ולא כ"קרובים" אליו, אין צורך שכל אברי הסדרה יהיו בסביבת הגבול כדי לענות להגדרה זו. כלומר, מספיק ש<b>כמעט כל האיברים</b> יהיו בסביבת הגבול, החל ממקום מסוים שאותו נהוג לסמן באות <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"> <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/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> גדולה או <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"> <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/5b6328fbe0cded37216c90735c89ee188be26a30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.509ex;" alt="{\displaystyle N_{0}}"></span>. על כך יורחב בהמשך. </p> <div class="mw-heading mw-heading3"><h3 id="המשמעות_של_"התקרבות_לגבול""><span id=".D7.94.D7.9E.D7.A9.D7.9E.D7.A2.D7.95.D7.AA_.D7.A9.D7.9C_.22.D7.94.D7.AA.D7.A7.D7.A8.D7.91.D7.95.D7.AA_.D7.9C.D7.92.D7.91.D7.95.D7.9C.22"></span>המשמעות של "התקרבות לגבול"</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&action=edit&section=4" title="עריכת קוד המקור של הפרק: המשמעות של "התקרבות לגבול""><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&veaction=edit&section=4" title="עריכת פסקה: "המשמעות של "התקרבות לגבול""" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>הגדרה מדויקת של מושג הגבול תדרוש כי עבור כל סביבה של הגבול, ניתן למצוא מקום מסוים המסומן באות <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"> <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/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> גדולה או <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"> <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/5b6328fbe0cded37216c90735c89ee188be26a30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.509ex;" alt="{\displaystyle N_{0}}"></span>, שהחל ממנו כל אברי הסדרה מצויים בתוך סביבה זו. דהיינו, עבור כל מרחק "קטן כרצוננו" מהגבול, קיים <a href="/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%98%D7%91%D7%A2%D7%99" title="מספר טבעי">מספר טבעי</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 N_{0}=N_{0}(\varepsilon )\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{0}=N_{0}(\varepsilon )\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c90f17e323aae60b22b5e70935d979d945faa3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.351ex; height:2.843ex;" alt="{\displaystyle N_{0}=N_{0}(\varepsilon )\in \mathbb {N} }"></span> (שיכול להיות גדול מאוד, ותלוי בבחירה 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 \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></span>), כך שכל איברי הסדרה מעבר לאותו מספר נמצאים בתוך מרחק זה מהגבול (הסימן של "קיים" הוא <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77ed842b6b90b2fdd825320cf8e5265fa937b583" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \exists }"></span>). כיוון 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\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d55d9b32f6fa8fab6a84ea444a6b5a24bb45e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\to \infty }"></span> עד <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"> <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/5b6328fbe0cded37216c90735c89ee188be26a30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.509ex;" alt="{\displaystyle N_{0}}"></span> יש רק מספר סופי של אינדקסים וממנו והלאה יש מספר אינסופי של אינדקסים. לכן אפשר לומר שעבור כל <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 \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></span>, <a href="/wiki/%D7%9B%D7%9E%D7%A2%D7%98_%D7%9B%D7%9C_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="כמעט כל (מתמטיקה)">כמעט כל</a> אברי הסדרה נמצאים במרחק שקטן 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 \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></span> מהגבול - כלומר לא משנה עד כמה נצמצם את הסביבה של הגבול, עדיין כמעט כל הסדרה תישאר בתוך אותה סביבה. </p> <dl><dd><b>הגדרה</b>: תהא <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 \left\{a_{n}\right\}_{n=1}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow> <mo>{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{a_{n}\right\}_{n=1}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac1970085b595e60458aa67afdd248ec563b9422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.092ex; height:3.176ex;" alt="{\displaystyle \left\{a_{n}\right\}_{n=1}^{\infty }}"></span> <a href="/wiki/%D7%A1%D7%93%D7%A8%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="סדרה (מתמטיקה)">סדרה</a> של <a href="/wiki/%D7%A9%D7%93%D7%94_%D7%94%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D_%D7%94%D7%9E%D7%9E%D7%A9%D7%99%D7%99%D7%9D" title="שדה המספרים הממשיים">מספרים ממשיים</a>. נאמר על הסדרה שהיא <b>מתכנסת</b> למספר הממשי <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 =L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e20be9e871bda5035225352ff389d0b13158140b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.036ex; height:2.176ex;" alt="{\displaystyle =L}"></span>, או 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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> הוא <b>הגבול</b> של הסדרה, ונסמן זאת <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 }a_{n}=L}"> <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">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }a_{n}=L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec8a6da951ff8db5ca56f8e9454ed61271cf6ffd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.789ex; height:3.676ex;" alt="{\displaystyle \lim _{n\to \infty }a_{n}=L}"></span> או בקיצור <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 a_{n}\to L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}\to L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11b96c0ee27d23b06518212eb970a21743f13383" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.645ex; height:2.509ex;" alt="{\displaystyle a_{n}\to L}"></span> אם לכל מספר ממשי <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 \varepsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04ec3670b50384a3ce48aca42e7cc5131a06b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}"></span> (קטן כרצוננו) קיים <a href="/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%98%D7%91%D7%A2%D7%99" title="מספר טבעי">מספר טבעי</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 N_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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/5b6328fbe0cded37216c90735c89ee188be26a30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.509ex;" 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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> המקיים <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>N_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>N_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9828ac419884804764040b66506b2bf38d903001" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.414ex; height:2.509ex;" alt="{\displaystyle n>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 \left|a_{n}-L\right|<\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>L</mi> </mrow> <mo>|</mo> </mrow> <mo><</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|a_{n}-L\right|<\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/115cbf6ce4bb333b0b0b1d0b24f55ebd84d96d4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.347ex; height:2.843ex;" alt="{\displaystyle \left|a_{n}-L\right|<\varepsilon }"></span>.</dd></dl> <p>צורת כתיבה נוספת היא: </p> <dl><dd><dl><dd><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 L=\lim _{n\to \infty }a_{n}\iff \forall \varepsilon >0,\,\exists N_{0}>0,\,\forall n>N_{0},\,|a_{n}-L|<\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mi>ε</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">∃</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">∀</mi> <mi>n</mi> <mo>></mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=\lim _{n\to \infty }a_{n}\iff \forall \varepsilon >0,\,\exists N_{0}>0,\,\forall n>N_{0},\,|a_{n}-L|<\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/726cc1330901e96762639000aa911c0cd07e0d15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:59.114ex; height:3.843ex;" alt="{\displaystyle L=\lim _{n\to \infty }a_{n}\iff \forall \varepsilon >0,\,\exists N_{0}>0,\,\forall n>N_{0},\,|a_{n}-L|<\varepsilon }"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading4"><h4 id="איתור_המספר_'"`UNIQ--postMath-00000034-QINU`"'_שממנו_והלאה_אברי_הסדרה_נמצאים_בסביבת_הגבול"><span id=".D7.90.D7.99.D7.AA.D7.95.D7.A8_.D7.94.D7.9E.D7.A1.D7.A4.D7.A8_.7F.27.22.60UNIQ--postMath-00000034-QINU.60.22.27.7F_.D7.A9.D7.9E.D7.9E.D7.A0.D7.95_.D7.95.D7.94.D7.9C.D7.90.D7.94_.D7.90.D7.91.D7.A8.D7.99_.D7.94.D7.A1.D7.93.D7.A8.D7.94_.D7.A0.D7.9E.D7.A6.D7.90.D7.99.D7.9D_.D7.91.D7.A1.D7.91.D7.99.D7.91.D7.AA_.D7.94.D7.92.D7.91.D7.95.D7.9C"></span>איתור המספר <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"> <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/5b6328fbe0cded37216c90735c89ee188be26a30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.509ex;" alt="{\displaystyle N_{0}}"></span> שממנו והלאה אברי הסדרה נמצאים בסביבת הגבול</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&action=edit&section=5" title="עריכת קוד המקור של הפרק: איתור המספר '"`UNIQ--postMath-00000034-QINU`"' שממנו והלאה אברי הסדרה נמצאים בסביבת הגבול"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&veaction=edit&section=5" title="עריכת פסקה: "איתור המספר '"`UNIQ--postMath-00000034-QINU`"' שממנו והלאה אברי הסדרה נמצאים בסביבת הגבול"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>יש לשים לב שהאינדקס <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"> <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/5b6328fbe0cded37216c90735c89ee188be26a30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.509ex;" alt="{\displaystyle N_{0}}"></span> תלוי ב -<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 \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></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 \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></span> יהיה קטן יותר, <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"> <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/5b6328fbe0cded37216c90735c89ee188be26a30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.509ex;" alt="{\displaystyle N_{0}}"></span> המתאים לו, עשוי להיות גדול יותר. לעיתים מסמנים <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_{\varepsilon }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{\varepsilon }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c8e08441e5dd8cff849c4a84d9852848d1975c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.865ex; height:2.509ex;" alt="{\displaystyle N_{\varepsilon }}"></span> או <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}(\varepsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{0}(\varepsilon )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e7f871e37552cb37fcbdfaaa81b50f162c9814" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.813ex; height:2.843ex;" alt="{\displaystyle N_{0}(\varepsilon )}"></span> במקום <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"> <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/5b6328fbe0cded37216c90735c89ee188be26a30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.509ex;" alt="{\displaystyle N_{0}}"></span> כדי להדגיש עובדה זו. </p><p>כמו כן, יש לזכור שהאינדקס <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"> <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/5b6328fbe0cded37216c90735c89ee188be26a30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.509ex;" alt="{\displaystyle N_{0}}"></span> הוא מספר טבעי (<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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{0}\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23c87f8d941e45dc2d00aeec00019c228b902ced" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.439ex; height:2.509ex;" alt="{\displaystyle N_{0}\in \mathbb {N} }"></span>). כלומר, על פי הגדרת הגבול עליו להיות <a href="/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%A9%D7%9C%D7%9D" title="מספר שלם">מספר שלם</a> חיובי. לעיתים במהלך החישוב או ההוכחה של גבול של סדרה מסוימת יתקבל <a href="/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%9E%D7%9E%D7%A9%D7%99" title="מספר ממשי">מספר ממשי</a> חיובי שאינו שלם. במקרה זה האינדקס <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"> <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/5b6328fbe0cded37216c90735c89ee188be26a30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.509ex;" alt="{\displaystyle N_{0}}"></span> יהיה <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%99%D7%AA_%D7%94%D7%A2%D7%A8%D7%9A_%D7%94%D7%A9%D7%9C%D7%9D" title="פונקציית הערך השלם">הערך השלם העליון</a> של המספר שהתקבל בחישוב. שכן משום שהוא גדול יותר מהמספר שהתקבל, גם ממנו והלאה אברי הסדרה יתכנסו לעבר הגבול. כדי להדגיש שמדובר בערך העליון (פונקציית התקרה) הוא יסומן כך: <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 \lceil N\rceil }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌈</mo> <mi>N</mi> <mo fence="false" stretchy="false">⌉</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lceil N\rceil }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba5d1b1092649ddda8bdaf1e775d41e525e9461a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.128ex; height:2.843ex;" alt="{\displaystyle \lceil N\rceil }"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="אפיון_התכנסות_לפי_קושי"><span id=".D7.90.D7.A4.D7.99.D7.95.D7.9F_.D7.94.D7.AA.D7.9B.D7.A0.D7.A1.D7.95.D7.AA_.D7.9C.D7.A4.D7.99_.D7.A7.D7.95.D7.A9.D7.99"></span>אפיון התכנסות לפי קושי</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&action=edit&section=6" title="עריכת קוד המקור של הפרק: אפיון התכנסות לפי קושי"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&veaction=edit&section=6" title="עריכת פסקה: "אפיון התכנסות לפי קושי"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>תנאי קושי</b> הוא אפיון שקול לכך שסדרה ממשית מתכנסת. סדרה המקיימת את תנאי קושי היא סדרה שהמרחק בין כל שני איברים שגדולים מאינדקס כלשהו, קטן כרצוננו. ניתן לשים לב שאף על פי שסדרה המקיימת את תנאי קושי בהכרח מתכנסת לגבול סופי, בהגדרה הפורמלית של תנאי קושי לא מופיע כלל ערך הגבול אליו הסדרה מתכנסת, ומכאן גם חשיבותו של אפיון זה: הוא מספק את האפשרות לקבוע האם סדרה מתכנסת מבלי להתייחס לגבול אליו היא מתכנסת, בניגוד להגדרת הגבול שמחייבת התייחסות לערך הגבול. </p> <dl><dd><b>הגדרה</b>: תהא <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 \left\{a_{n}\right\}_{n=1}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow> <mo>{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{a_{n}\right\}_{n=1}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac1970085b595e60458aa67afdd248ec563b9422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.092ex; height:3.176ex;" alt="{\displaystyle \left\{a_{n}\right\}_{n=1}^{\infty }}"></span> סדרה של <a href="/wiki/%D7%A9%D7%93%D7%94_%D7%94%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D_%D7%94%D7%9E%D7%9E%D7%A9%D7%99%D7%99%D7%9D" title="שדה המספרים הממשיים">מספרים ממשיים</a>. נאמר על הסדרה שהיא מקיימת את תנאי קושי אם לכל מספר ממשי <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 \varepsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04ec3670b50384a3ce48aca42e7cc5131a06b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}"></span> (קטן כרצוננו) קיים <a href="/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%98%D7%91%D7%A2%D7%99" title="מספר טבעי">מספר טבעי</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 N_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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/5b6328fbe0cded37216c90735c89ee188be26a30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.509ex;" 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 m,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6568e95b6bf8f39b7fd2c9b52b7b00ee124c6250" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.469ex; height:2.009ex;" alt="{\displaystyle m,n}"></span> המקיימים <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 m,n>N_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>></mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m,n>N_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86b13dc1d134c4c7cdf713ad85982c38f2812460" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.488ex; height:2.509ex;" alt="{\displaystyle m,n>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 \left|a_{n}-a_{m}\right|<\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mo><</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|a_{n}-a_{m}\right|<\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/803306da080faacc3cc80740e348c998a2a9d922" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.669ex; height:2.843ex;" alt="{\displaystyle \left|a_{n}-a_{m}\right|<\varepsilon }"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="גבול_במובן_הרחב"><span id=".D7.92.D7.91.D7.95.D7.9C_.D7.91.D7.9E.D7.95.D7.91.D7.9F_.D7.94.D7.A8.D7.97.D7.91"></span>גבול במובן הרחב</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&action=edit&section=7" title="עריכת קוד המקור של הפרק: גבול במובן הרחב"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&veaction=edit&section=7" title="עריכת פסקה: "גבול במובן הרחב"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>אומרים שסדרה <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 a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span> <b>שואפת לאינסוף</b> (או שאינסוף הוא גבול הסדרה), אם לכל מספר ממשי <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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> קיים <a href="/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%98%D7%91%D7%A2%D7%99" title="מספר טבעי">מספר טבעי</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 N_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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/5b6328fbe0cded37216c90735c89ee188be26a30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.509ex;" 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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> המקיים <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>N_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>N_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9828ac419884804764040b66506b2bf38d903001" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.414ex; height:2.509ex;" alt="{\displaystyle n>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 a_{n}>M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>></mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}>M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8acfce8e35184d9a57e4ccb39436f03726ad747e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.989ex; height:2.509ex;" alt="{\displaystyle a_{n}>M}"></span>. ההגדרה של <b>שאיפה למינוס אינסוף</b> דומה. </p><p>כך למשל סדרה 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 1,2,3,4,5,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,2,3,4,5,\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7accecae06e727f166cf52ff97eddbaf6e7195a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.705ex; height:2.509ex;" alt="{\displaystyle 1,2,3,4,5,\ldots }"></span> שואפת לאינסוף, כי ה"זנב" שלה גדול מכל מספר ממשי שנרצה, אולם הסדרה <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 1,2,1,3,1,4\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,2,1,3,1,4\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0dbd0dc20746f3df8b036e8a6eadda45107c197" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.255ex; height:2.509ex;" alt="{\displaystyle 1,2,1,3,1,4\ldots }"></span> אינה שואפת לאינסוף, כי היא אמנם גדולה כרצוננו, אולם לא כל איברי ה"זנב" גדולים כרצוננו. </p> <div class="mw-heading mw-heading2"><h2 id="הגבול_כאופרטור"><span id=".D7.94.D7.92.D7.91.D7.95.D7.9C_.D7.9B.D7.90.D7.95.D7.A4.D7.A8.D7.98.D7.95.D7.A8"></span>הגבול כאופרטור</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&action=edit&section=8" title="עריכת קוד המקור של הפרק: הגבול כאופרטור"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&veaction=edit&section=8" title="עריכת פסקה: "הגבול כאופרטור"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>בתוך <a href="/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99" title="מרחב וקטורי">המרחב הווקטורי</a> של כל הסדרות הממשיות, שאותו מסמנים ב-<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 \mathbb {R} ^{\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{\mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2608c429068c69675f1144f0ba8249603a8ceaf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.097ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{\mathbb {N} }}"></span> או <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 \mathbb {R} ^{\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{\omega }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0881ff53d61be4b3ed806e41af0d2ffba29c09b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.933ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{\omega }}"></span>, אוסף הסדרות המתכנסות מהווה <a href="/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)" title="אלגברה (מבנה אלגברי)">אלגברה</a>, שעליה מוגדר <a href="/wiki/%D7%90%D7%95%D7%A4%D7%A8%D7%98%D7%95%D7%A8" title="אופרטור">אופרטור</a> הגבול: האופרטור מחזיר, עבור סדרה מתכנסת <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 a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span>, את גבולה <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 }a_{n}}"> <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">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2a80ff6955b824f704802ffc03d3fe59270d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.108ex; height:3.676ex;" alt="{\displaystyle \lim _{n\to \infty }a_{n}}"></span>, שהוא מספר ממשי. </p><p>לפי <a href="/wiki/%D7%90%D7%A8%D7%99%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94_%D7%A9%D7%9C_%D7%92%D7%91%D7%95%D7%9C%D7%95%D7%AA" title="אריתמטיקה של גבולות">אריתמטיקה של גבולות</a>, בהינתן שתי סדרות מתכנסות <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 a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></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 b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28e2d72f6dd9375c8f1f59f1effd9b4e5492ac97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.509ex;" alt="{\displaystyle b_{n}}"></span> וקבוע <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> מתקיים </p> <div dir="ltr" class="mw-content-ltr"> <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 \ \lim _{n\to \infty }(a_{n}+b_{n})=\lim _{n\to \infty }a_{n}+\lim _{n\to \infty }b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \lim _{n\to \infty }(a_{n}+b_{n})=\lim _{n\to \infty }a_{n}+\lim _{n\to \infty }b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cd1fc57449afdf94391908120e384817242e6ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.477ex; height:3.843ex;" alt="{\displaystyle \ \lim _{n\to \infty }(a_{n}+b_{n})=\lim _{n\to \infty }a_{n}+\lim _{n\to \infty }b_{n}}"></span></li> <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 \ \lim _{n\to \infty }(a_{n}\cdot b_{n})=\lim _{n\to \infty }a_{n}\cdot \lim _{n\to \infty }b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \lim _{n\to \infty }(a_{n}\cdot b_{n})=\lim _{n\to \infty }a_{n}\cdot \lim _{n\to \infty }b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/119e96760d92b8f2bd659d66eee815f7725377bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.154ex; height:3.843ex;" alt="{\displaystyle \ \lim _{n\to \infty }(a_{n}\cdot b_{n})=\lim _{n\to \infty }a_{n}\cdot \lim _{n\to \infty }b_{n}}"></span></li> <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 \ \lim _{n\to \infty }(\alpha \cdot b_{n})=\alpha \cdot \lim _{n\to \infty }b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo stretchy="false">(</mo> <mi>α</mi> <mo>⋅</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>α</mi> <mo>⋅</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \lim _{n\to \infty }(\alpha \cdot b_{n})=\alpha \cdot \lim _{n\to \infty }b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c66c7738f23b666c9735c7f31b43dfc0f5a436ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.573ex; height:3.843ex;" alt="{\displaystyle \ \lim _{n\to \infty }(\alpha \cdot b_{n})=\alpha \cdot \lim _{n\to \infty }b_{n}}"></span></li> <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 \ \lim _{n\to \infty }\left({\frac {a_{n}}{b_{n}}}\right)={\frac {\lim _{n\to \infty }a_{n}}{\lim _{n\to \infty }b_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \lim _{n\to \infty }\left({\frac {a_{n}}{b_{n}}}\right)={\frac {\lim _{n\to \infty }a_{n}}{\lim _{n\to \infty }b_{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c396247b2e574f3c55a5b2ff0093d41b860e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.837ex; height:6.176ex;" alt="{\displaystyle \ \lim _{n\to \infty }\left({\frac {a_{n}}{b_{n}}}\right)={\frac {\lim _{n\to \infty }a_{n}}{\lim _{n\to \infty }b_{n}}}}"></span></li></ul> </div> <p>כאשר בסעיף האחרון נדרש 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 \lim _{n\to \infty }b_{n}\neq 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">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }b_{n}\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebc03e5cfbce2316f7baa360bb110477ff1f8d0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.137ex; height:3.676ex;" alt="{\displaystyle \lim _{n\to \infty }b_{n}\neq 0}"></span> וש-<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 b_{n}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d90ff22063787d2ae0dd938ffdf2aa2eacd7626" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.477ex; height:2.676ex;" alt="{\displaystyle b_{n}\neq 0}"></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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="מושגים_קרובים"><span id=".D7.9E.D7.95.D7.A9.D7.92.D7.99.D7.9D_.D7.A7.D7.A8.D7.95.D7.91.D7.99.D7.9D"></span>מושגים קרובים</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&action=edit&section=9" title="עריכת קוד המקור של הפרק: מושגים קרובים"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&veaction=edit&section=9" title="עריכת פסקה: "מושגים קרובים"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>כאשר סדרה אינה מתכנסת, אין לה גבול במובן שהוגדר למעלה, ואז נדרשים כלים מעט אחרים. ההרחבה הטבעית הראשונה היא לסדרות שגבולן אינסוף או מינוס אינסוף, והן "<b>מתכנסות במובן הרחב</b>", כפי שהוסבר לעיל. </p><p>גבול של <a href="/wiki/%D7%AA%D7%AA-%D7%A1%D7%93%D7%A8%D7%94" title="תת-סדרה">תת-סדרה</a> נקרא <b>גבול חלקי</b> של הסדרה המקורית. כאשר סדרה מתכנסת כל הגבולות החלקיים שלה שווים. אוסף כל הגבולות החלקיים מתאר במובן ידוע את הסדרה המקורית; הגבול החלקי הקטן ביותר נקרא <b>גבול תחתון</b>, והגדול ביותר הוא <b>גבול עליון</b>. </p> <div class="mw-heading mw-heading2"><h2 id="ראו_גם"><span id=".D7.A8.D7.90.D7.95_.D7.92.D7.9D"></span>ראו גם</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&action=edit&section=10" title="עריכת קוד המקור של הפרק: ראו גם"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&veaction=edit&section=10" title="עריכת פסקה: "ראו גם"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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)">גבול (מתמטיקה)</a></li> <li><a href="/wiki/%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94" title="גבול של פונקציה">גבול של פונקציה</a></li> <li><a href="/wiki/%D7%92%D7%91%D7%95%D7%9C_(%D7%98%D7%95%D7%A4%D7%95%D7%9C%D7%95%D7%92%D7%99%D7%94)" title="גבול (טופולוגיה)">גבול (טופולוגיה)</a></li> <li><a href="/wiki/%D7%9E%D7%91%D7%97%D7%A0%D7%99_%D7%94%D7%AA%D7%9B%D7%A0%D7%A1%D7%95%D7%AA_%D7%9C%D7%A1%D7%93%D7%A8%D7%95%D7%AA" title="מבחני התכנסות לסדרות">מבחני התכנסות לסדרות</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><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&action=edit&section=11" title="עריכת קוד המקור של הפרק: קישורים חיצוניים"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94&veaction=edit&section=11" title="עריכת פסקה: "קישורים חיצוניים"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="sistersitebox plainlinks noprint" style="margin: 0 1em 0.5em 0;float: left;"><tbody><tr><th style="text-align:center">מיזמי <a href="/wiki/%D7%A7%D7%A8%D7%9F_%D7%95%D7%99%D7%A7%D7%99%D7%9E%D7%93%D7%99%D7%94" title="קרן ויקימדיה">קרן ויקימדיה</a></th></tr><tr><td><div class="sisterwikilinkT"><span typeof="mw:File"><span title="ויקיספר"><img alt="ויקיספר" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/20px-Wikibooks-logo.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/30px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/40px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></span></span> ספר לימוד בוויקיספר: <b><a href="https://he.wikibooks.org/wiki/%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%90%D7%99%D7%A0%D7%A4%D7%99%D7%A0%D7%99%D7%98%D7%A1%D7%99%D7%9E%D7%9C%D7%99/%D7%92%D7%91%D7%95%D7%9C%D7%95%D7%AA" class="extiw" title="b:חשבון אינפיניטסימלי/גבולות">חשבון אינפיניטסימלי/גבולות</a></b></div></td></tr><tr><td><div class="sisterwikilinkT"><span typeof="mw:File"><span title="ויקישיתוף"><img alt="ויקישיתוף" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/20px-Commons-logo.svg.png" decoding="async" width="20" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> תמונות ומדיה בוויקישיתוף: <b><a href="https://commons.wikimedia.org/wiki/Category:Limit_of_a_sequence" class="extiw" title="commons:Category:Limit of a sequence">גבול של סדרה</a></b></div></td></tr></tbody></table><style data-mw-deduplicate="TemplateStyles:r36773993">.mw-parser-output .sistersitebox{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9}</style> <p><br /> </p> <table class="navbox nowraplinks mw-collapsible autocollapse" 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%A0%D7%9C%D7%99%D7%96%D7%94_%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%AA_-_%D7%9E%D7%95%D7%A0%D7%97%D7%99%D7%9D" title="אנליזה מתמטית - מונחים">חשבון אינפיניטסימלי</a> </th></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">מושגי יסוד </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%90%D7%99%D7%A0%D7%A4%D7%99%D7%A0%D7%99%D7%98%D7%A1%D7%99%D7%9E%D7%9C%D7%99" title="חשבון אינפיניטסימלי">חשבון אינפיניטסימלי</a> • <a href="/wiki/%D7%A1%D7%93%D7%A8%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="סדרה (מתמטיקה)">סדרה</a> • <a href="/wiki/%D7%A1%D7%93%D7%A8%D7%94_%D7%9E%D7%AA%D7%9B%D7%A0%D7%A1%D7%AA" title="סדרה מתכנסת">סדרה מתכנסת</a> • <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)">גבול</a> • <a href="/wiki/%D7%A1%D7%93%D7%A8%D7%AA_%D7%A7%D7%95%D7%A9%D7%99" title="סדרת קושי">סדרת קושי</a> • <a href="/wiki/%D7%98%D7%95%D7%A8_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="טור (מתמטיקה)">טור</a> • <a href="/wiki/%D7%90%D7%99%D7%A0%D7%A4%D7%99%D7%A0%D7%99%D7%98%D7%A1%D7%99%D7%9E%D7%9C" title="אינפיניטסימל">אינפיניטסימל</a> • <a href="/wiki/%D7%A9%D7%93%D7%94_%D7%94%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D_%D7%94%D7%9E%D7%9E%D7%A9%D7%99%D7%99%D7%9D" title="שדה המספרים הממשיים">שדה המספרים הממשיים</a> • <a href="/wiki/%D7%A2%D7%A8%D7%9A_%D7%9E%D7%95%D7%97%D7%9C%D7%98" title="ערך מוחלט">ערך מוחלט</a> • <a href="/wiki/%D7%90%D7%99-%D7%A9%D7%95%D7%95%D7%99%D7%95%D7%9F_%D7%94%D7%9E%D7%A9%D7%95%D7%9C%D7%A9" title="אי-שוויון המשולש">אי-שוויון המשולש</a> • <a href="/wiki/%D7%90%D7%99-%D7%A9%D7%95%D7%95%D7%99%D7%95%D7%9F_%D7%A7%D7%95%D7%A9%D7%99-%D7%A9%D7%95%D7%95%D7%A8%D7%A5" title="אי-שוויון קושי-שוורץ">אי-שוויון קושי-שוורץ</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">פונקציות </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%9E%D7%9E%D7%A9%D7%99%D7%AA" title="פונקציה ממשית">פונקציה</a> • <a href="/wiki/%D7%92%D7%A8%D7%A3_%D7%A9%D7%9C_%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94" title="גרף של פונקציה">גרף פונקציה</a> • <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="פונקציה ליניארית">פונקציה ליניארית</a> • <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%9E%D7%95%D7%A0%D7%95%D7%98%D7%95%D7%A0%D7%99%D7%AA" title="פונקציה מונוטונית">פונקציה מונוטונית</a> • <a href="/wiki/%D7%A0%D7%A7%D7%95%D7%93%D7%AA_%D7%A7%D7%99%D7%A6%D7%95%D7%9F" title="נקודת קיצון">נקודת קיצון</a> •<a href="/wiki/%D7%A0%D7%A7%D7%95%D7%93%D7%AA_%D7%A4%D7%99%D7%AA%D7%95%D7%9C" title="נקודת פיתול">נקודת פיתול</a> •<a href="/wiki/%D7%A0%D7%A7%D7%95%D7%93%D7%AA_%D7%90%D7%95%D7%9B%D7%A3" title="נקודת אוכף">נקודת אוכף</a> • <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%A7%D7%A2%D7%95%D7%A8%D7%94" title="פונקציה קעורה">פונקציה קעורה</a> • <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%A7%D7%9E%D7%95%D7%A8%D7%94" title="פונקציה קמורה">פונקציה קמורה</a> • <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%A8%D7%A6%D7%99%D7%A4%D7%94_(%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94)" title="פונקציה רציפה (אנליזה)">פונקציה רציפה</a> • <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%A8%D7%A6%D7%99%D7%A4%D7%94_%D7%91%D7%9E%D7%99%D7%93%D7%94_%D7%A9%D7%95%D7%95%D7%94" title="פונקציה רציפה במידה שווה">פונקציה רציפה במידה שווה</a> • <a href="/wiki/%D7%A0%D7%A7%D7%95%D7%93%D7%AA_%D7%90%D7%99_%D7%A8%D7%A6%D7%99%D7%A4%D7%95%D7%AA" title="נקודת אי רציפות">נקודת אי רציפות</a> • <a href="/wiki/%D7%A0%D7%92%D7%96%D7%A8%D7%AA" title="נגזרת">נגזרת</a> • <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> • <a href="/wiki/%D7%A1%D7%93%D7%A8%D7%AA_%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%95%D7%AA" title="סדרת פונקציות">סדרת פונקציות</a> • <a href="/wiki/%D7%94%D7%AA%D7%9B%D7%A0%D7%A1%D7%95%D7%AA_%D7%A0%D7%A7%D7%95%D7%93%D7%AA%D7%99%D7%AA" title="התכנסות נקודתית">התכנסות נקודתית</a> • <a href="/wiki/%D7%94%D7%AA%D7%9B%D7%A0%D7%A1%D7%95%D7%AA_%D7%91%D7%9E%D7%99%D7%93%D7%94_%D7%A9%D7%95%D7%95%D7%94" title="התכנסות במידה שווה">התכנסות במידה שווה</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">משפטים </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%91%D7%95%D7%9C%D7%A6%D7%90%D7%A0%D7%95-%D7%95%D7%99%D7%99%D7%A8%D7%A9%D7%98%D7%A8%D7%90%D7%A1" title="משפט בולצאנו-ויירשטראס">משפט בולצאנו-ויירשטראס</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98%D7%99_%D7%95%D7%99%D7%99%D7%A8%D7%A9%D7%98%D7%A8%D7%90%D7%A1" title="משפטי ויירשטראס">משפטי ויירשטראס</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%A7%D7%A0%D7%98%D7%95%D7%A8_%D7%9C%D7%A8%D7%A6%D7%99%D7%A4%D7%95%D7%AA_%D7%91%D7%9E%D7%99%D7%93%D7%94_%D7%A9%D7%95%D7%95%D7%94" title="משפט קנטור לרציפות במידה שווה">משפט קנטור</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%A2%D7%A8%D7%9A_%D7%94%D7%91%D7%99%D7%A0%D7%99%D7%99%D7%9D" title="משפט ערך הביניים">משפט ערך הביניים</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%A4%D7%A8%D7%9E%D7%94_(%D7%9C%D7%A0%D7%A7%D7%95%D7%93%D7%95%D7%AA_%D7%A7%D7%99%D7%A6%D7%95%D7%9F)" title="משפט פרמה (לנקודות קיצון)">משפט פרמה</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%A8%D7%95%D7%9C" title="משפט רול">משפט רול</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%A2%D7%A8%D7%9A_%D7%94%D7%9E%D7%9E%D7%95%D7%A6%D7%A2_%D7%A9%D7%9C_%D7%9C%D7%92%D7%A8%D7%90%D7%A0%D7%96%27" title="משפט הערך הממוצע של לגראנז'">משפט הערך הממוצע של לגראנז'</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%A2%D7%A8%D7%9A_%D7%94%D7%9E%D7%9E%D7%95%D7%A6%D7%A2_%D7%A9%D7%9C_%D7%A7%D7%95%D7%A9%D7%99" title="משפט הערך הממוצע של קושי">משפט הערך הממוצע של קושי</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%93%D7%90%D7%A8%D7%91%D7%95" title="משפט דארבו">משפט דארבו</a> • <a href="/wiki/%D7%9B%D7%9C%D7%9C_%D7%94%D7%A9%D7%A8%D7%A9%D7%A8%D7%AA" title="כלל השרשרת">כלל השרשרת</a> • <a href="/wiki/%D7%9B%D7%9C%D7%9C_%D7%94%D7%A1%D7%A0%D7%93%D7%95%D7%95%D7%99%D7%A5%27" title="כלל הסנדוויץ'">כלל הסנדוויץ'</a> • <a href="/wiki/%D7%9B%D7%9C%D7%9C_%D7%9C%D7%95%D7%A4%D7%99%D7%98%D7%9C" title="כלל לופיטל">כלל לופיטל</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%A9%D7%98%D7%95%D7%9C%D7%A5" title="משפט שטולץ">משפט שטולץ</a> • <a href="/wiki/%D7%90%D7%A8%D7%99%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94_%D7%A9%D7%9C_%D7%92%D7%91%D7%95%D7%9C%D7%95%D7%AA" title="אריתמטיקה של גבולות">אריתמטיקה של גבולות</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">האינטגרל </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C" title="אינטגרל">אינטגרל</a> • <a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%9C%D7%90_%D7%90%D7%9E%D7%99%D7%AA%D7%99" title="אינטגרל לא אמיתי">אינטגרל לא אמיתי</a> • <a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%A8%D7%91-%D7%9E%D7%9E%D7%93%D7%99" title="אינטגרל רב-ממדי">אינטגרל רב-ממדי</a> • <a href="/wiki/%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99" title="המשפט היסודי של החשבון הדיפרנציאלי והאינטגרלי">המשפט היסודי של החשבון הדיפרנציאלי והאינטגרלי</a> • <a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%A6%D7%99%D7%94_%D7%91%D7%97%D7%9C%D7%A7%D7%99%D7%9D" title="אינטגרציה בחלקים">אינטגרציה בחלקים</a> • <a href="/wiki/%D7%A9%D7%99%D7%98%D7%95%D7%AA_%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%A6%D7%99%D7%94" title="שיטות אינטגרציה">שיטות אינטגרציה</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">אנליזה מתקדמת </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%9E%D7%A8%D7%95%D7%9B%D7%91%D7%AA" title="פונקציה מרוכבת">פונקציה מרוכבת</a> • <a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99%D7%AA" title="אנליזה וקטורית">אנליזה וקטורית</a> • <a href="/wiki/%D7%A9%D7%99%D7%98%D7%AA_%D7%A0%D7%99%D7%95%D7%98%D7%95%D7%9F-%D7%A8%D7%A4%D7%A1%D7%95%D7%9F" title="שיטת ניוטון-רפסון">שיטת ניוטון-רפסון</a> • <a href="/wiki/%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%94_%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99%D7%AA" title="משוואה דיפרנציאלית">משוואה דיפרנציאלית</a> • <a href="/wiki/%D7%98%D7%95%D7%A4%D7%95%D7%9C%D7%95%D7%92%D7%99%D7%94" title="טופולוגיה">טופולוגיה</a> • <a href="/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%9E%D7%99%D7%93%D7%94" title="תורת המידה">תורת המידה</a> </td></tr> <tr> <td colspan="3" style="background-color: #F2F3F4; text-align: center; font-weight: bold;"><a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%AA" title="אנליזה מתמטית">אנליזה מתמטית</a> • <a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99%D7%AA" title="אנליזה וקטורית">אנליזה וקטורית</a> • <a href="/wiki/%D7%98%D7%95%D7%A4%D7%95%D7%9C%D7%95%D7%92%D7%99%D7%94" title="טופולוגיה">טופולוגיה</a> • <a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%9E%D7%A8%D7%95%D7%9B%D7%91%D7%AA" title="אנליזה מרוכבת">אנליזה מרוכבת</a> • <a 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