CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="sk" dir="ltr"> <head> <meta charset="UTF-8"> <title>Limita – Wikipédia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )skwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":[",\t."," \t,"],"wgDigitTransformTable":["",""], "wgDefaultDateFormat":"dmy","wgMonthNames":["","január","február","marec","apríl","máj","jún","júl","august","september","október","november","december"],"wgRequestId":"5413b0f2-8d84-4a2f-b20a-7a611dda22c8","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Limita","wgTitle":"Limita","wgCurRevisionId":7953653,"wgRevisionId":7953653,"wgArticleId":95856,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Matematická analýza","Vlastnosti matematických funkcií"],"wgPageViewLanguage":"sk","wgPageContentLanguage":"sk","wgPageContentModel":"wikitext","wgRelevantPageName":"Limita","wgRelevantArticleId":95856,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0, "wgVisualEditor":{"pageLanguageCode":"sk","pageLanguageDir":"ltr","pageVariantFallbacks":"sk"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":7000,"wgRelatedArticlesCompat":[],"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q177239","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false,"wgSiteNoticeId":"2.2"};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading", "ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready","ext.dismissableSiteNotice.styles":"ready"};RLPAGEMODULES=["site","mediawiki.page.ready","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.WikiMiniAtlas","ext.gadget.edit-summaries","ext.urlShortener.toolbar","ext.centralauth.centralautologin","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession" ,"ext.dismissableSiteNotice"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=sk&modules=ext.dismissableSiteNotice.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=sk&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=sk&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.6"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Limita – Wikipédia"> <meta property="og:type" content="website"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//sk.m.wikipedia.org/wiki/Limita"> <link rel="alternate" type="application/x-wiki" title="Upraviť" href="/w/index.php?title=Limita&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipédia (sk)"> <link rel="EditURI" type="application/rsd+xml" href="//sk.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://sk.wikipedia.org/wiki/Limita"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.sk"> <link rel="alternate" type="application/atom+xml" title="Atom kanál Wikipédia" href="/w/index.php?title=%C5%A0peci%C3%A1lne:Posledn%C3%A9%C3%9Apravy&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Limita rootpage-Limita skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Preskočiť na obsah</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Projekt"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Hlavné menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Hlavné menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Hlavné menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">presunúť do postranného panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">skryť</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigácia </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage" class="mw-list-item"><a href="/wiki/Hlavn%C3%A1_str%C3%A1nka" title="Navštíviť Hlavnú stránku [z]" accesskey="z"><span>Hlavná stránka</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikip%C3%A9dia:Port%C3%A1l_komunity" title="O projekte, ako môžete prispieť, kde čo nájsť"><span>Portál komunity</span></a></li><li id="n-kaviaren" class="mw-list-item"><a href="/wiki/Wikip%C3%A9dia:Kaviare%C5%88"><span>Kaviareň</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/%C5%A0peci%C3%A1lne:Posledn%C3%A9%C3%9Apravy" title="Zoznam posledných úprav na tejto wiki [r]" accesskey="r"><span>Posledné úpravy</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/%C5%A0peci%C3%A1lne:N%C3%A1hodn%C3%A1" title="Zobraziť náhodnú stránku [x]" accesskey="x"><span>Náhodná stránka</span></a></li><li id="n-help" class="mw-list-item"><a href="https://sk.wikipedia.org/wiki/Pomoc:Obsah" title="Miesto, kde nájdete pomoc"><span>Pomocník</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Hlavn%C3%A1_str%C3%A1nka" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipédia" src="/static/images/mobile/copyright/wikipedia-wordmark-fr.svg" style="width: 7.4375em; height: 1.125em;"> <img class="mw-logo-tagline" alt="Slobodná encyklopédia" src="/static/images/mobile/copyright/wikipedia-tagline-sk.svg" width="118" height="13" style="width: 7.375em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/%C5%A0peci%C3%A1lne:H%C4%BEadanie" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Vyhľadať na Wikipédia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Hľadať</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Hľadať na Wikipédii" aria-label="Hľadať na Wikipédii" autocapitalize="sentences" title="Vyhľadať na Wikipédia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Špeciálne:Hľadanie"> </div> <button class="cdx-button cdx-search-input__end-button">Hľadať</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Osobné nástroje"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Vzhľad"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Zmeniť vzhľad veľkosti písma, šírky stránky a farby" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Vzhľad" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Vzhľad</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=sk.wikipedia.org&uselang=sk" class=""><span>Prispieť</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%C5%A0peci%C3%A1lne:Vytvorenie%C3%9A%C4%8Dtu&returnto=Limita" title="Odporúčame vytvoriť si vlastný účet a prihlásiť sa, nie je to však povinné." class=""><span>Vytvoriť účet</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%C5%A0peci%C3%A1lne:Prihl%C3%A1seniePou%C5%BE%C3%ADvate%C4%BEa&returnto=Limita" title="Odporúčame vám prihlásiť sa, nie je to však povinné. [o]" accesskey="o" class=""><span>Prihlásiť sa</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Ďalšie možnosti" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Osobné nástroje" > <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">Osobné nástroje</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Uživatelské menu" > <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="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=sk.wikipedia.org&uselang=sk"><span>Prispieť</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%C5%A0peci%C3%A1lne:Vytvorenie%C3%9A%C4%8Dtu&returnto=Limita" title="Odporúčame vytvoriť si vlastný účet a prihlásiť sa, nie je to však povinné."><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Vytvoriť účet</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%C5%A0peci%C3%A1lne:Prihl%C3%A1seniePou%C5%BE%C3%ADvate%C4%BEa&returnto=Limita" title="Odporúčame vám prihlásiť sa, nie je to však povinné. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Prihlásiť sa</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"> Stránky pre odhlásených redaktorov <a href="/wiki/Pomoc:%C3%9Avod" aria-label="Viac informácií o editovaní"><span>zistiť viac</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/%C5%A0peci%C3%A1lne:MojePr%C3%ADspevky" title="Zoznam úprav vykonaných z tejto IP adresy [y]" accesskey="y"><span>Príspevky</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%C5%A0peci%C3%A1lne:MojaDiskusia" title="Diskusia o úpravách z tejto ip adresy [n]" accesskey="n"><span>Diskusia</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\"\u003Enezobrazovať\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=\"sk\" dir=\"ltr\"\u003E\u003Ctable width=\"100%\" class=\"skin-nightmode-reset-color\" style=\"background:var( --background-color-base, #FFF); color:#27408B;\"\u003E\n\n\u003Ctbody\u003E\u003Ctr valign=\"top\"\u003E\n\u003Ctd style=\"text-align:center;\"\u003E\u003Cdiv style=\"display: flex; text-align:center; border: 1px solid black;\" class=\"plainlinks\"\u003E\u003Cdiv style=\"padding-left:0.5em; padding-top:0.2em; margin-bottom: -0.4em;\"\u003E\u003Cfigure class=\"mw-halign-left\" typeof=\"mw:File\"\u003E\u003Ca href=\"/wiki/Wikip%C3%A9dia:%C3%81zijsk%C3%BD_mesiac_Wikip%C3%A9die\" title=\"Ázijský mesiac Wikipédie\"\u003E\u003Cimg alt=\"Ázijský mesiac Wikipédie\" src=\"//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/WAM_logo_without_text.svg/38px-WAM_logo_without_text.svg.png\" decoding=\"async\" width=\"38\" height=\"41\" class=\"mw-file-element\" srcset=\"//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/WAM_logo_without_text.svg/57px-WAM_logo_without_text.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/WAM_logo_without_text.svg/76px-WAM_logo_without_text.svg.png 2x\" data-file-width=\"800\" data-file-height=\"859\" /\u003E\u003C/a\u003E\u003Cfigcaption\u003EÁzijský mesiac Wikipédie\u003C/figcaption\u003E\u003C/figure\u003E\u003C/div\u003E\u003Cdiv style=\"text-align:center; flex: 1; margin: 0.2em 0;\"\u003E\u003Cbig\u003E\u003Cb\u003E\u003Ca href=\"/wiki/Wikip%C3%A9dia:%C3%81zijsk%C3%BD_mesiac_Wikip%C3%A9die_2024\" title=\"Wikipédia:Ázijský mesiac Wikipédie 2024\"\u003EÁzijský mesiac Wikipédie 2024\u003C/a\u003E\u003C/b\u003E\u003C/big\u003E\u003Cbr /\u003E\u003Cb\u003EZapojte sa a\u0026#160;získajte cenu\u003C/b\u003E\n\u003C/div\u003E\n\u003C/div\u003E\u003C/td\u003E\u003C/tr\u003E\u003C/tbody\u003E\u003C/table\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E";}}());</script></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Projekt"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Obsah" 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">Obsah</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">presunúť do postranného panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">skryť</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">Začiatok</div> </a> </li> <li id="toc-Limita_funkcie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Limita_funkcie"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Limita funkcie</span> </div> </a> <button aria-controls="toc-Limita_funkcie-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>Přepnout podsekci Limita funkcie</span> </button> <ul id="toc-Limita_funkcie-sublist" class="vector-toc-list"> <li id="toc-Formálna_definícia" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formálna_definícia"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Formálna definícia</span> </div> </a> <ul id="toc-Formálna_definícia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limita_funkcie_v_nekonečne" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Limita_funkcie_v_nekonečne"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Limita funkcie v nekonečne</span> </div> </a> <ul id="toc-Limita_funkcie_v_nekonečne-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Limita_postupnosti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Limita_postupnosti"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Limita postupnosti</span> </div> </a> <ul id="toc-Limita_postupnosti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Externé_odkazy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Externé_odkazy"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Externé odkazy</span> </div> </a> <ul id="toc-Externé_odkazy-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="Obsah" 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="Prepnúť obsah" > <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">Prepnúť obsah</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">Limita</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="Prejsť na článok v inom jazyku. Je dostupný v 79 jazykoch" > <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-79" 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">79 jazykov</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Limiet" title="Limiet – afrikánčina" lang="af" hreflang="af" data-title="Limiet" data-language-autonym="Afrikaans" data-language-local-name="afrikánčina" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8C%A5%E1%8C%8D" title="ጥግ – amharčina" lang="am" hreflang="am" data-title="ጥግ" data-language-autonym="አማርኛ" data-language-local-name="amharčina" class="interlanguage-link-target"><span>አማርኛ</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_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="نهاية (رياضيات) – arabčina" lang="ar" hreflang="ar" data-title="نهاية (رياضيات)" data-language-autonym="العربية" data-language-local-name="arabčina" 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_matem%C3%A1tica" title="Llende matemática – astúrčina" lang="ast" hreflang="ast" data-title="Llende matemática" data-language-autonym="Asturianu" data-language-local-name="astúrčina" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Limit_(riyaziyyat)" title="Limit (riyaziyyat) – azerbajdžančina" lang="az" hreflang="az" data-title="Limit (riyaziyyat)" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajdžančina" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BA%D0%BB%D3%99%D0%BC%D3%99_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Сикләмә (математика) – baškirčina" lang="ba" hreflang="ba" data-title="Сикләмә (математика)" data-language-autonym="Башҡортса" data-language-local-name="baškirčina" 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%BC%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D0%BA%D0%B0)" title="Ліміт (матэматыка) – bieloruština" lang="be" hreflang="be" data-title="Ліміт (матэматыка)" data-language-autonym="Беларуская" data-language-local-name="bieloruština" 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%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Граница (математика) – bulharčina" lang="bg" hreflang="bg" data-title="Граница (математика)" data-language-autonym="Български" data-language-local-name="bulharčina" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%B8%E0%A5%80%E0%A4%AE%E0%A4%BE_(%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4)" title="सीमा (गणित) – Bhojpuri" lang="bh" hreflang="bh" data-title="सीमा (गणित)" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A7%80%E0%A6%AE%E0%A6%BE_(%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4)" title="সীমা (গণিত) – bengálčina" lang="bn" hreflang="bn" data-title="সীমা (গণিত)" data-language-autonym="বাংলা" data-language-local-name="bengálčina" 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/Limes_(matematika)" title="Limes (matematika) – bosniačtina" lang="bs" hreflang="bs" data-title="Limes (matematika)" data-language-autonym="Bosanski" data-language-local-name="bosniačtina" 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/L%C3%ADmit" title="Límit – katalánčina" lang="ca" hreflang="ca" data-title="Límit" data-language-autonym="Català" data-language-local-name="katalánčina" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%95%D8%A7%D8%AF%DB%95_(%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9)" title="ڕادە (ماتماتیک) – kurdčina (sorání)" lang="ckb" hreflang="ckb" data-title="ڕادە (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="kurdčina (sorání)" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Limita" title="Limita – čeština" lang="cs" hreflang="cs" data-title="Limita" data-language-autonym="Čeština" data-language-local-name="čeština" 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%A7%D0%B8%D0%BA%C4%95_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Чикĕ (математика) – čuvaština" lang="cv" hreflang="cv" data-title="Чикĕ (математика)" data-language-autonym="Чӑвашла" data-language-local-name="čuvaština" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Terfan_(mathemateg)" title="Terfan (mathemateg) – waleština" lang="cy" hreflang="cy" data-title="Terfan (mathemateg)" data-language-autonym="Cymraeg" data-language-local-name="waleština" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Gr%C3%A6nsev%C3%A6rdi_(matematik)" title="Grænseværdi (matematik) – dánčina" lang="da" hreflang="da" data-title="Grænseværdi (matematik)" data-language-autonym="Dansk" data-language-local-name="dánčina" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Limit" title="Limit – Zazaki" lang="diq" hreflang="diq" data-title="Limit" data-language-autonym="Zazaki" data-language-local-name="Zazaki" class="interlanguage-link-target"><span>Zazaki</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%BC%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC)" title="Όριο (μαθηματικά) – gréčtina" lang="el" hreflang="el" data-title="Όριο (μαθηματικά)" data-language-autonym="Ελληνικά" data-language-local-name="gréčtina" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Limit_(mathematics)" title="Limit (mathematics) – angličtina" lang="en" hreflang="en" data-title="Limit (mathematics)" data-language-autonym="English" data-language-local-name="angličtina" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Limeso" title="Limeso – esperanto" lang="eo" hreflang="eo" data-title="Limeso" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/L%C3%ADmite_(matem%C3%A1tica)" title="Límite (matemática) – španielčina" lang="es" hreflang="es" data-title="Límite (matemática)" data-language-autonym="Español" data-language-local-name="španielčina" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Limite" title="Limite – baskičtina" lang="eu" hreflang="eu" data-title="Limite" data-language-autonym="Euskara" data-language-local-name="baskičtina" 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%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="حد (ریاضی) – perzština" lang="fa" hreflang="fa" data-title="حد (ریاضی)" data-language-autonym="فارسی" data-language-local-name="perzština" 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/Raja-arvo" title="Raja-arvo – fínčina" lang="fi" hreflang="fi" data-title="Raja-arvo" data-language-autonym="Suomi" data-language-local-name="fínčina" 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_(math%C3%A9matiques)" title="Limite (mathématiques) – francúzština" lang="fr" hreflang="fr" data-title="Limite (mathématiques)" data-language-autonym="Français" data-language-local-name="francúzština" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Teorainn_(matamaitic)" title="Teorainn (matamaitic) – írčina" lang="ga" hreflang="ga" data-title="Teorainn (matamaitic)" data-language-autonym="Gaeilge" data-language-local-name="írčina" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E6%A5%B5%E9%99%90" title="極限 – čínština (dialekty Gan)" lang="gan" hreflang="gan" data-title="極限" data-language-autonym="贛語" data-language-local-name="čínština (dialekty Gan)" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Limit_(mat%C3%A9matik)" title="Limit (matématik) – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Limit (matématik)" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/L%C3%ADmite_matem%C3%A1tico" title="Límite matemático – galícijčina" lang="gl" hreflang="gl" data-title="Límite matemático" data-language-autonym="Galego" data-language-local-name="galícijčina" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-hak mw-list-item"><a href="https://hak.wikipedia.org/wiki/Khi%CC%8Dt-han_(S%C3%BA-ho%CC%8Dk)" title="Khi̍t-han (Sú-ho̍k) – čínština (dialekty Hakka)" lang="hak" hreflang="hak" data-title="Khi̍t-han (Sú-ho̍k)" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="čínština (dialekty Hakka)" class="interlanguage-link-target"><span>客家語 / Hak-kâ-ngî</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/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) – hebrejčina" lang="he" hreflang="he" data-title="גבול (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="hebrejčina" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A5%80%E0%A4%AE%E0%A4%BE_(%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4)" title="सीमा (गणित) – hindčina" lang="hi" hreflang="hi" data-title="सीमा (गणित)" data-language-autonym="हिन्दी" data-language-local-name="hindčina" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Limes_(matematika)" title="Limes (matematika) – chorvátčina" lang="hr" hreflang="hr" data-title="Limes (matematika)" data-language-autonym="Hrvatski" data-language-local-name="chorvátčina" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Hat%C3%A1r%C3%A9rt%C3%A9k" title="Határérték – maďarčina" lang="hu" hreflang="hu" data-title="Határérték" data-language-autonym="Magyar" data-language-local-name="maďarčina" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8D%D5%A1%D5%B0%D5%B4%D5%A1%D5%B6_(%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Սահման (մաթեմատիկա) – arménčina" lang="hy" hreflang="hy" data-title="Սահման (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="arménčina" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Limit_(matematika)" title="Limit (matematika) – indonézština" lang="id" hreflang="id" data-title="Limit (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonézština" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Limito" title="Limito – ido" lang="io" hreflang="io" data-title="Limito" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Markgildi" title="Markgildi – islandčina" lang="is" hreflang="is" data-title="Markgildi" data-language-autonym="Íslenska" data-language-local-name="islandčina" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Limite_(matematica)" title="Limite (matematica) – taliančina" lang="it" hreflang="it" data-title="Limite (matematica)" data-language-autonym="Italiano" data-language-local-name="taliančina" 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%A5%B5%E9%99%90" title="極限 – japončina" lang="ja" hreflang="ja" data-title="極限" data-language-autonym="日本語" data-language-local-name="japončina" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Limit_(matimatix)" title="Limit (matimatix) – jamajská kreolština" lang="jam" hreflang="jam" data-title="Limit (matimatix)" data-language-autonym="Patois" data-language-local-name="jamajská kreolština" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-ka badge-Q70893996 mw-list-item" title=""><a href="https://ka.wikipedia.org/wiki/%E1%83%96%E1%83%A6%E1%83%95%E1%83%90%E1%83%A0%E1%83%98_(%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%90)" title="ზღვარი (მათემატიკა) – gruzínčina" lang="ka" hreflang="ka" data-title="ზღვარი (მათემატიკა)" data-language-autonym="ქართული" data-language-local-name="gruzínčina" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Talast_(Tusnakt)" title="Talast (Tusnakt) – kabylčina" lang="kab" hreflang="kab" data-title="Talast (Tusnakt)" data-language-autonym="Taqbaylit" data-language-local-name="kabylčina" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A8%D0%B5%D0%BA" title="Шек – kazaština" lang="kk" hreflang="kk" data-title="Шек" data-language-autonym="Қазақша" data-language-local-name="kazaština" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%9B%E1%9E%B8%E1%9E%98%E1%9E%B8%E1%9E%8F" title="លីមីត – khmérčina" lang="km" hreflang="km" data-title="លីមីត" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khmérčina" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AA%E0%B2%B0%E0%B2%BF%E0%B2%AE%E0%B2%BF%E0%B2%A4%E0%B2%BF" title="ಪರಿಮಿತಿ – kannadčina" lang="kn" hreflang="kn" data-title="ಪರಿಮಿತಿ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannadčina" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B7%B9%ED%95%9C" title="극한 – kórejčina" lang="ko" hreflang="ko" data-title="극한" data-language-autonym="한국어" data-language-local-name="kórejčina" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Limes_(mathematica)" title="Limes (mathematica) – latinčina" lang="la" hreflang="la" data-title="Limes (mathematica)" data-language-autonym="Latina" data-language-local-name="latinčina" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo badge-Q17437796 badge-featuredarticle mw-list-item" title="najlepší článok"><a href="https://lmo.wikipedia.org/wiki/Limit_(matematega)" title="Limit (matematega) – lombardština" lang="lmo" hreflang="lmo" data-title="Limit (matematega)" data-language-autonym="Lombard" data-language-local-name="lombardština" 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/Riba_(matematika)" title="Riba (matematika) – litovčina" lang="lt" hreflang="lt" data-title="Riba (matematika)" data-language-autonym="Lietuvių" data-language-local-name="litovčina" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Robe%C5%BEa_(matem%C4%81tika)" title="Robeža (matemātika) – lotyština" lang="lv" hreflang="lv" data-title="Robeža (matemātika)" data-language-autonym="Latviešu" data-language-local-name="lotyština" class="interlanguage-link-target"><span>Latviešu</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" title="Гранична вредност – macedónčina" lang="mk" hreflang="mk" data-title="Гранична вредност" data-language-autonym="Македонски" data-language-local-name="macedónčina" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B8%E0%A5%80%E0%A4%AE%E0%A4%BE_(%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4)" title="सीमा (गणित) – maráthčina" lang="mr" hreflang="mr" data-title="सीमा (गणित)" data-language-autonym="मराठी" data-language-local-name="maráthčina" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Had_(matematik)" title="Had (matematik) – malajčina" lang="ms" hreflang="ms" data-title="Had (matematik)" data-language-autonym="Bahasa Melayu" data-language-local-name="malajčina" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Limiet" title="Limiet – holandčina" lang="nl" hreflang="nl" data-title="Limiet" data-language-autonym="Nederlands" data-language-local-name="holandčina" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Grense_i_matematikk" title="Grense i matematikk – nórčina (nynorsk)" lang="nn" hreflang="nn" data-title="Grense i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="nórčina (nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Grenseverdi" title="Grenseverdi – nórčina (bokmal)" lang="nb" hreflang="nb" data-title="Grenseverdi" data-language-autonym="Norsk bokmål" data-language-local-name="nórčina (bokmal)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Granica_(matematyka)" title="Granica (matematyka) – poľština" lang="pl" hreflang="pl" data-title="Granica (matematyka)" data-language-autonym="Polski" data-language-local-name="poľština" 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" title="Limite – portugalčina" lang="pt" hreflang="pt" data-title="Limite" data-language-autonym="Português" data-language-local-name="portugalčina" 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_(matematic%C4%83)" title="Limită (matematică) – rumunčina" lang="ro" hreflang="ro" data-title="Limită (matematică)" data-language-autonym="Română" data-language-local-name="rumunčina" 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%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Предел (математика) – ruština" lang="ru" hreflang="ru" data-title="Предел (математика)" data-language-autonym="Русский" data-language-local-name="ruština" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/L%C3%ACmiti" title="Lìmiti – sicílčina" lang="scn" hreflang="scn" data-title="Lìmiti" data-language-autonym="Sicilianu" data-language-local-name="sicílčina" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Limes_(matematika)" title="Limes (matematika) – srbochorvátčina" lang="sh" hreflang="sh" data-title="Limes (matematika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srbochorvátčina" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Limit_(mathematics)" title="Limit (mathematics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Limit (mathematics)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Limiti" title="Limiti – albánčina" lang="sq" hreflang="sq" data-title="Limiti" data-language-autonym="Shqip" data-language-local-name="albánčina" class="interlanguage-link-target"><span>Shqip</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" title="Гранична вредност – srbčina" lang="sr" hreflang="sr" data-title="Гранична вредност" data-language-autonym="Српски / srpski" data-language-local-name="srbčina" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Gr%C3%A4nsv%C3%A4rde" title="Gränsvärde – švédčina" lang="sv" hreflang="sv" data-title="Gränsvärde" data-language-autonym="Svenska" data-language-local-name="švédčina" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%8E%E0%AE%B2%E0%AF%8D%E0%AE%B2%E0%AF%88_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="எல்லை (கணிதம்) – tamilčina" lang="ta" hreflang="ta" data-title="எல்லை (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="tamilčina" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Limit" title="Limit – turečtina" lang="tr" hreflang="tr" data-title="Limit" data-language-autonym="Türkçe" data-language-local-name="turečtina" 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" title="Границя – ukrajinčina" lang="uk" hreflang="uk" data-title="Границя" data-language-autonym="Українська" data-language-local-name="ukrajinčina" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AD%D8%AF_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="حد (ریاضی) – urdčina" lang="ur" hreflang="ur" data-title="حد (ریاضی)" data-language-autonym="اردو" data-language-local-name="urdčina" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Limit" title="Limit – uzbečtina" lang="uz" hreflang="uz" data-title="Limit" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbečtina" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Limite" title="Limite – benátština" lang="vec" hreflang="vec" data-title="Limite" data-language-autonym="Vèneto" data-language-local-name="benátština" class="interlanguage-link-target"><span>Vèneto</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_(to%C3%A1n_h%E1%BB%8Dc)" title="Giới hạn (toán học) – vietnamčina" lang="vi" hreflang="vi" data-title="Giới hạn (toán học)" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamčina" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Limiet" title="Limiet – vlámština (západní)" lang="vls" hreflang="vls" data-title="Limiet" data-language-autonym="West-Vlams" data-language-local-name="vlámština (západní)" class="interlanguage-link-target"><span>West-Vlams</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%9E%81%E9%99%90%EF%BC%88%E6%95%B0%E5%AD%A6%EF%BC%89" title="极限（数学） – čínština (wu)" lang="wuu" hreflang="wuu" data-title="极限（数学）" data-language-autonym="吴语" data-language-local-name="čínština (wu)" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%9E%81%E9%99%90_(%E6%95%B0%E5%AD%A6)" title="极限 (数学) – čínština" lang="zh" hreflang="zh" data-title="极限 (数学)" data-language-autonym="中文" data-language-local-name="čínština" 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%AD%B8)" title="極限 (數學) – kantončina" lang="yue" hreflang="yue" data-title="極限 (數學)" data-language-autonym="粵語" data-language-local-name="kantončina" 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/Q177239#sitelinks-wikipedia" title="Upraviť medzijazykové odkazy" class="wbc-editpage">Upraviť odkazy</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="Menné priestory"> <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/Limita" title="Zobraziť obsah stránky [c]" accesskey="c"><span>Stránka</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Diskusia:Limita&action=edit&redlink=1" rel="discussion" class="new" title="Diskusia o obsahu stránky (stránka neexistuje) [t]" accesskey="t"><span>Diskusia</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="Změnit variantu jazyka" > <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">slovenčina</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="Zobrazenia"> <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/Limita"><span>Čítať</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Limita&veaction=edit" title="Upraviť túto stránku [v]" accesskey="v"><span>Upraviť</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Limita&action=edit" title="Upraviť zdrojový kód tejto stránky [e]" accesskey="e"><span>Upraviť zdroj</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Limita&action=history" title="Minulé revízie tejto stránky. [h]" accesskey="h"><span>Zobraziť históriu</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Nástroje stránky"> <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="Nástroje" > <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">Nástroje</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">Nástroje</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">presunúť do postranného panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">skryť</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Další možnosti" > <div class="vector-menu-heading"> Akcie </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/Limita"><span>Čítať</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Limita&veaction=edit" title="Upraviť túto stránku [v]" accesskey="v"><span>Upraviť</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Limita&action=edit" title="Upraviť zdrojový kód tejto stránky [e]" accesskey="e"><span>Upraviť zdroj</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Limita&action=history"><span>Zobraziť históriu</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Všeobecné </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%C5%A0peci%C3%A1lne:%C4%8CoOdkazujeSem/Limita" title="Zoznam všetkých wiki stránok, ktoré sem odkazujú [j]" accesskey="j"><span>Odkazy na túto stránku</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%C5%A0peci%C3%A1lne:S%C3%BAvisiacePosledn%C3%A9%C3%9Apravy/Limita" rel="nofollow" title="Posledné úpravy na stránkach, na ktoré odkazuje táto stránka [k]" accesskey="k"><span>Súvisiace úpravy</span></a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=sk" title="Nahranie súborov [u]" accesskey="u"><span>Nahrať súbor</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%C5%A0peci%C3%A1lne:%C5%A0peci%C3%A1lneStr%C3%A1nky" title="Zoznam všetkých špeciálnych stránok [q]" accesskey="q"><span>Špeciálne stránky</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Limita&oldid=7953653" title="Trvalý odkaz na túto verziu stránky"><span>Trvalý odkaz</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Limita&action=info" title="Viac informácií o tejto stránke"><span>Informácie o stránke</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=%C5%A0peci%C3%A1lne:Citova%C5%A5&page=Limita&id=7953653&wpFormIdentifier=titleform" title="Informácie ako citovať túto stránku"><span>Citovať túto stránku</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=%C5%A0peci%C3%A1lne:UrlQ%C4%B1sald%C4%B1c%C4%B1s%C4%B1&url=https%3A%2F%2Fsk.wikipedia.org%2Fwiki%2FLimita"><span>Získať skrátené URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=%C5%A0peci%C3%A1lne:QrKodu&url=https%3A%2F%2Fsk.wikipedia.org%2Fwiki%2FLimita"><span>Stiahnuť QR kód</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"> Tlačiť/exportovať </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=%C5%A0peci%C3%A1lne:Knihy&bookcmd=book_creator&referer=Limita"><span>Vytvoriť knihu</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%C5%A0peci%C3%A1lne:DownloadAsPdf&page=Limita&action=show-download-screen"><span>Stiahnuť ako PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Limita&printable=yes" title="Verzia tejto stránky pre tlač [p]" accesskey="p"><span>Verzia pre tlač</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"> V iných projektoch </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_(mathematics)" hreflang="en"><span>Wikimedia Commons</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/Q177239" title="Odkaz na prepojenú položku dátového úložiska [g]" accesskey="g"><span>Položka Wikidata</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="Nástroje stránky"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Vzhľad"> <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">Vzhľad</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">presunúť do postranného panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">skryť</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">z Wikipédie, slobodnej encyklopédie</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-ltr mw-parser-output" lang="sk" dir="ltr"><p><b>Limita</b> je v matematike hodnota, ku ktorej sa "približuje" premenlivá hodnota. </p><p>Limita funkcie sa používa na opis správania sa <a href="/wiki/Zobrazenie_(matematika)" title="Zobrazenie (matematika)">funkcie</a>, keď sa jej <a href="/wiki/Argument" class="mw-disambig" title="Argument">argument</a> "približuje" k nejakému bodu alebo rastie do <a href="/wiki/Nekone%C4%8Dno_(matematika)" title="Nekonečno (matematika)">nekonečna</a>. Limity sa využívajú v <a href="/wiki/Matematick%C3%A1_anal%C3%BDza" title="Matematická analýza">matematickej analýze</a> na definovanie <a href="/wiki/Deriv%C3%A1cia_(funkcia)" title="Derivácia (funkcia)">derivácie</a> a <a href="/wiki/Spojit%C3%A1_funkcia" title="Spojitá funkcia">spojitosti</a>. </p><p>Koncept "limity funkcie" sa dá ďalej zovšeobecniť na <a href="/w/index.php?title=Topologick%C3%A1_sie%C5%A5&action=edit&redlink=1" class="new" title="Topologická sieť (stránka neexistuje)">topologickú sieť</a> a limita postupnosti úzko spojená s <a href="/w/index.php?title=Limita_(te%C3%B3ria_kateg%C3%B3ri%C3%AD)&action=edit&redlink=1" class="new" title="Limita (teória kategórií) (stránka neexistuje)">limitou</a> a <a href="/w/index.php?title=Priama_limita&action=edit&redlink=1" class="new" title="Priama limita (stránka neexistuje)">priamou limitou</a> v <a href="/w/index.php?title=Te%C3%B3ria_kateg%C3%B3ri%C3%AD&action=edit&redlink=1" class="new" title="Teória kategórií (stránka neexistuje)">teórii kategórií</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Limita_funkcie">Limita funkcie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limita&veaction=edit&section=1" title="Upraviť sekciu: Limita funkcie" class="mw-editsection-visualeditor"><span>upraviť</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Limita&action=edit&section=1" title="Editovat zdrojový kód sekce Limita funkcie"><span>upraviť zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nech <i>f</i> je reálna funkcia a <i>c</i> je reálne číslo. Výraz: </p> <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 \lim _{x\to c}f(x)=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>x</mi> <mo stretchy="false">→</mo> <mi>c</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to c}f(x)=L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3f174c16cc9e61932ecc7eb71b27a4f10a44fc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.781ex; height:3.843ex;" alt="{\displaystyle \lim _{x\to c}f(x)=L}"></span></dd></dl> <p>znamená, že hodnota <i>f</i>(<i>x</i>) sa dá ľubovoľne priblížiť k <i>L</i> tak, že sa <i>x</i> dostane dostatočne blízko k <i>c</i>. Potom hovoríme, že "limita funkcie <i>f</i> v bode <i>c</i> (alebo ako sa <i>f</i> približuje k <i>c</i>) je <i>L</i>". Limita môže existovať dokonca aj keď <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(c)\neq L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(c)\neq L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18baf4dce5b2c8d7c02c4e85c022105952910004" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.776ex; height:2.843ex;" alt="{\displaystyle f(c)\neq L}"></span>. Funkcia <i>f</i> nemusí byť ani definovaná v bode <i>c</i>. </p><p>Uvažujme <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {x}{x^{2}+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {x}{x^{2}+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0554a7a2bf1a57581bb8d432f437dc46f49f679" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.739ex; height:5.176ex;" alt="{\displaystyle f(x)={\frac {x}{x^{2}+1}}}"></span> ako sa <i>x</i> približuje k 2. V tomto prípade je funkcia <i>f</i> definovaná v bode 2 a jej hodnota 0,4 sa rovná limite v tomto bode: </p> <table class="wikitable"> <tbody><tr> <td>f(1.9)</td> <td>f(1.99)</td> <td>f(1.999)</td> <td>f(2)</td> <td>f(2.001)</td> <td>f(2.01)</td> <td>f(2.1) </td></tr> <tr> <td>0.4121</td> <td>0.4012</td> <td>0.4001</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" 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 \Rightarrow }"></span> 0.4 <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 \Leftarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇐</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/682eb97b10e06ba3d2dcc642ecd753d34dbb4ef9" 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 \Leftarrow }"></span></td> <td>0.3998</td> <td>0.3988</td> <td>0.3882 </td></tr></tbody></table> <p>Ako sa <i>x</i> približuje k 2, hodnoty <i>f</i>(<i>x</i>) sa približujú k 0,4 a preto môžeme napísať <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to 2}f(x)=0.4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mn>2</mn> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to 2}f(x)=0.4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7483bc0d539823fa3eff0e2b1f59b2616b666d8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.28ex; height:4.009ex;" alt="{\displaystyle \lim _{x\to 2}f(x)=0.4}"></span>. </p><p>Ale nie vždy je to také jednoduché. </p><p>Uvažujme </p> <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 g(x)=\left\{{\begin{matrix}{\frac {x}{x^{2}+1}},&{\mbox{ak }}x\neq 2\\\\0,&{\mbox{ak }}x=2.\end{matrix}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>ak </mtext> </mstyle> </mrow> <mi>x</mi> <mo>≠</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>ak </mtext> </mstyle> </mrow> <mi>x</mi> <mo>=</mo> <mn>2.</mn> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)=\left\{{\begin{matrix}{\frac {x}{x^{2}+1}},&{\mbox{ak }}x\neq 2\\\\0,&{\mbox{ak }}x=2.\end{matrix}}\right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47030843a1e66923164ee0a33ce5227d5caa19f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:27.057ex; height:10.176ex;" alt="{\displaystyle g(x)=\left\{{\begin{matrix}{\frac {x}{x^{2}+1}},&{\mbox{ak }}x\neq 2\\\\0,&{\mbox{ak }}x=2.\end{matrix}}\right.}"></span></dd></dl> <p>Limita funkcie <i>g</i> ako sa <i>x</i> približuje k 2 je 0,4 (tak ako v prípade funkcie <i>f</i>), ale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to 2}g(x)\neq g(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mn>2</mn> </mrow> </munder> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to 2}g(x)\neq g(2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693faf91e7606e02a21023440f4d8e577dc64fa5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.234ex; height:4.009ex;" alt="{\displaystyle \lim _{x\to 2}g(x)\neq g(2)}"></span>; funkcia <i>g</i> nie je spojitá v bode <i>x</i> = 2. </p><p>Alebo majme ešte funkciu <i>f</i>, ktorá je nedefinovaná v bode <i>x</i> = <i>c</i>. </p> <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 f(x)={\frac {x-1}{{\sqrt {x}}-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {x-1}{{\sqrt {x}}-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/141a9e1090bc2badb51c673ae6f9fbc19881a284" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:15.621ex; height:6.176ex;" alt="{\displaystyle f(x)={\frac {x-1}{{\sqrt {x}}-1}}}"></span></dd></dl> <p>V tomto prípade ako sa <i>x</i> približuje k 1, funkcia <i>f</i> je nedefinovaná v bode <i>x</i> = 1, ale jej limita v tomto bode existuje a rovná sa 2: </p> <table class="wikitable"> <tbody><tr> <td>f(0.9)</td> <td>f(0.99)</td> <td>f(0.999)</td> <td>f(1.0)</td> <td>f(1.001)</td> <td>f(1.01)</td> <td>f(1.1) </td></tr> <tr> <td>1.95</td> <td>1.99</td> <td>1.999</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" 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 \Rightarrow }"></span> nedef. <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 \Leftarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇐</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/682eb97b10e06ba3d2dcc642ecd753d34dbb4ef9" 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 \Leftarrow }"></span></td> <td>2.001</td> <td>2.010</td> <td>2.10 </td></tr></tbody></table> <p>Teda <i>x</i> sa môže približovať k 1, pokiaľ sa nebude rovnať 1, takže limita funkcie <i>f</i> v bode 1 je 2. </p> <div class="mw-heading mw-heading3"><h3 id="Formálna_definícia"><span id="Form.C3.A1lna_defin.C3.ADcia"></span>Formálna definícia</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limita&veaction=edit&section=2" title="Upraviť sekciu: Formálna definícia" class="mw-editsection-visualeditor"><span>upraviť</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Limita&action=edit&section=2" title="Editovat zdrojový kód sekce Formálna definícia"><span>upraviť zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Limitu formálne definujeme nasledovne: </p><p>Nech <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> je funkcia a bod <i>c</i> je <a href="/wiki/Hromadn%C3%BD_bod" title="Hromadný bod">hromadný bod</a> na jej <a href="/wiki/Defini%C4%8Dn%C3%BD_obor" title="Definičný obor">definičnom obore</a> (teda v každom <a href="/w/index.php?title=Prstencov%C3%A9_okolie&action=edit&redlink=1" class="new" title="Prstencové okolie (stránka neexistuje)">prstencovom okolí</a> bodu <i>c</i> sa nachádza aspoň jeden prvok z definičného oboru funkcie <i>f</i>). Nech <i>L</i> je reálne číslo. Potom zápis </p> <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 \lim _{x\to c}f(x)=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>x</mi> <mo stretchy="false">→</mo> <mi>c</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to c}f(x)=L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3f174c16cc9e61932ecc7eb71b27a4f10a44fc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.781ex; height:3.843ex;" alt="{\displaystyle \lim _{x\to c}f(x)=L}"></span></dd></dl> <p>znamená, že pre každé <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> <mtext> </mtext> <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/9c61af7583aa1e454a952bcfb120873b460b0575" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.925ex; height:2.176ex;" alt="{\displaystyle \varepsilon \ >0}"></span> existuje <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \ >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mtext> </mtext> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \ >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46a5cf1c76bec3924c737a75da7850fe1e2b74f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.89ex; height:2.343ex;" alt="{\displaystyle \delta \ >0}"></span> také, že pre všetky <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>, kde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<|x-c|<\delta \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>δ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<|x-c|<\delta \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c9ff0b0ed13bd4389098a64f996930d71107999" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.459ex; height:2.843ex;" alt="{\displaystyle 0<|x-c|<\delta \ }"></span>, platí <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f(x)-L|<\varepsilon \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(x)-L|<\varepsilon \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ebcaad1a5c7b00a224e64437843ffd97d6a6fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.897ex; height:2.843ex;" alt="{\displaystyle |f(x)-L|<\varepsilon \ }"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Limita_funkcie_v_nekonečne"><span id="Limita_funkcie_v_nekone.C4.8Dne"></span>Limita funkcie v nekonečne</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limita&veaction=edit&section=3" title="Upraviť sekciu: Limita funkcie v nekonečne" class="mw-editsection-visualeditor"><span>upraviť</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Limita&action=edit&section=3" title="Editovat zdrojový kód sekce Limita funkcie v nekonečne"><span>upraviť zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Limitu funkcie môžeme rozšíriť z prípadov približovania sa k reálnemu číslu na približovanie sa k <a href="/wiki/Nekone%C4%8Dno" title="Nekonečno">nekonečnu</a>. V týchto prípadoch toto približovanie sa k nekonečnu vlastne znamená, že <i>x</i> je stále väčšie a väčšie ("približuje" sa ku kladnému nekonečnu) alebo je stále menšie a menšie ("približuje" sa k zápornému nekonečnu). </p><p>Majme napríklad funkciu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {2x}{x+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {2x}{x+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/477d975e65fcfd55718757cdaee5d3c161cd10a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.685ex; height:5.343ex;" alt="{\displaystyle f(x)={\frac {2x}{x+1}}}"></span>. </p> <ul><li><i>f</i>(100) = 1.9802</li> <li><i>f</i>(1000) = 1.9980</li> <li><i>f</i>(10000) = 1.9998</li></ul> <p>Postupne ako sa <i>x</i> stáva veľmi veľkým, hodnota <i>f</i>(<i>x</i>) sa približuje k 2 a dá sa priblížiť ľubovoľne blízko k 2 výberom dostatočne veľkého <i>x</i>. V takomto prípade hovoríme, že limita funkcie <i>f</i> ako sa sa jej argument približuje k nekonečnu je 2. V matematickom zápise: </p> <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 \lim _{x\to \infty }f(x)=2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to \infty }f(x)=2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60de4f7cb63cc9ddb391f116c83f91729947e3dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.939ex; height:3.843ex;" alt="{\displaystyle \lim _{x\to \infty }f(x)=2.}"></span></dd></dl> <p>Formálne môžeme napísať túto definíciu: </p> <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 \lim _{x\to \infty }f(x)=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to \infty }f(x)=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5caffe5db5da84642da1ab2377d0e8eda8dad304" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.136ex; height:3.843ex;" alt="{\displaystyle \lim _{x\to \infty }f(x)=c}"></span> (definotoricky) len vtedy a vtedy, keď pre každé <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon >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 \epsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}"></span> existuje <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> také, že <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f(x)-c|<\epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(x)-c|<\epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6799f154951c3129dd1e19b1236e2400e115fea8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.601ex; height:2.843ex;" alt="{\displaystyle |f(x)-c|<\epsilon }"></span> pre všetky <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>n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>></mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x>n.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6bb56b187ca91f4c6cdfc4150545fd01ef56c9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.47ex; height:1.843ex;" alt="{\displaystyle x>n.}"></span></dd></dl> <p>Podobnú definíciu môžeme urobiť pre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to -\infty }f(x)=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to -\infty }f(x)=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/864a19d1cb367ac3c4b1420f3bf69d6170856f5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.415ex; height:4.009ex;" alt="{\displaystyle \lim _{x\to -\infty }f(x)=c}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Limita_postupnosti">Limita postupnosti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limita&veaction=edit&section=4" title="Upraviť sekciu: Limita postupnosti" class="mw-editsection-visualeditor"><span>upraviť</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Limita&action=edit&section=4" title="Editovat zdrojový kód sekce Limita postupnosti"><span>upraviť zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Uvažujme nasledujúcu postupnosť: 1,79; 1,799; 1,7999, ... Vidíme, že čísla sa postupne „približujú“ k číslu 1,8 -- limite postupnosti. </p><p>Formálne, predpokladajme, že <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub> je <a href="/wiki/Postupnos%C5%A5_(matematika)" title="Postupnosť (matematika)">postupnosť</a> <a href="/wiki/Re%C3%A1lne_%C4%8D%C3%ADslo" title="Reálne číslo">reálnych čísel</a>. Hovoríme, že reálne číslo <i>L</i> je <i>limita</i> tejto postupnosti a zapisujeme ako </p> <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 \lim _{n\to \infty }x_{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>x</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 }x_{n}=L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e98f0a690ec5ead63d484089b2d349613a708369" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.889ex; height:3.676ex;" alt="{\displaystyle \lim _{n\to \infty }x_{n}=L}"></span></dd></dl> <p>(definitoricky) len vtedy a vtedy, ak </p> <dl><dd>pre každé ε>0 existuje prirodzené číslo <i>n</i><sub>0</sub> také, že pre všetky <i>n</i>><i>n</i><sub>0</sub> platí |<i>x</i><sub><i>n</i></sub> - <i>L</i>| < ε.</dd></dl> <p>Intuitívne to znamená, že niekedy všetky prvky postupnosti sa dostanú tak blízko k limite, ako len budeme chcieť, keďže <a href="/wiki/Absol%C3%BAtna_hodnota_(re%C3%A1lne_a_komplexn%C3%A9_%C4%8D%C3%ADslo)" title="Absolútna hodnota (reálne a komplexné číslo)">absolútna hodnota</a> |<i>x</i><sub><i>n</i></sub> - <i>L</i>| sa dá interpretovať ako vzdialenosť <i>x</i><sub><i>n</i></sub> od <i>L</i>. Nie každá postupnosť má limitu; ak ju má, voláme ju <i>konvergentná</i>, inak je <i>divergentná</i>. Dá sa ukázať, že ak limita existuje, je jediná. </p><p>Limita postupnosti a limita funkcie sú, samozrejme, veľmi úzko späté. Postupnosť nie je nič iné ako funkcia definovaná na množine <a href="/wiki/Prirodzen%C3%A9_%C4%8D%C3%ADslo" title="Prirodzené číslo">prirodzených čísel</a>. Keďže jediným <a href="/wiki/Hromadn%C3%BD_bod" title="Hromadný bod">hromadným bodom</a> tejto množiny je kladné <a href="/wiki/Nekone%C4%8Dno" title="Nekonečno">nekonečno</a>, má zmysel skúmať limitu iba tu. Vyššie spomenutá definícia je len rozpísanie limity funkcie v nekonečne a ekvivalentne upravenej pre prirodzené čísla. Na druhej strane sa dá tento špeciálny prípad -- limita postupnosti -- použiť aj na definovanie limity funkcie, keďže limita funkcie <i>f</i> v bode <i>x</i> sa rovná limite postupnosti <i>x</i><sub><i>n</i></sub>=<i>f</i>(<i>x</i>+1/<i>n</i>) (ak existuje). </p><p>Vo všeobecnej situácii je ťažké ak nie nemožné zistiť či postupnosť má limitu tým, že sa overí definícia - teda že sa nájde limita. Ak uvažujeme o postupnosti reálnych čísel a o jej (prípadnej) reálnej limite, tak máme Cauchy-Bolzanove kritérium existencie limity, ktoré neoperuje samotnou hodnotou limity: postupnosť reálnych čísel <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"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </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/339d461b497fb8f8670bb29308fe09f0e7bfd34a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.773ex; height:2.843ex;" alt="{\displaystyle \{a_{n}\}}"></span> je konvergentná (t. j. má limitu) práve vtedy keď pre každé <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon >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 \epsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}"></span> existuje <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\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b985ba501f78cb9890f3ecda3e2e315cbd5cb26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.582ex; height:2.176ex;" alt="{\displaystyle N\in \mathbb {N} }"></span> že pre všetky <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 p,q>N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>></mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p,q>N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f35efa65081b931c629c7e1cbcd3b30223cd6721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.524ex; height:2.509ex;" alt="{\displaystyle p,q>N}"></span> je <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_{p}-a_{q}|<\epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a_{p}-a_{q}|<\epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aa050c5017377d5418dba6daf2475b347dffba9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.684ex; height:3.009ex;" alt="{\displaystyle |a_{p}-a_{q}|<\epsilon }"></span>. Slovne povedané: postupnosť má limitu práve vtedy keď pre každú presnosť <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon >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 \epsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}"></span> existuje také prirodzené číslo, že ak vyberieme ľubovoľne dva prvky z postupnosti ktorých poradové číslo je väčšie ako dotyčné prirodzené číslo, tak to už zaručí, že vybrané dva prvky postupnosti sú k sebe bližšie ako <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon >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 \epsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Externé_odkazy"><span id="Extern.C3.A9_odkazy"></span>Externé odkazy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limita&veaction=edit&section=5" title="Upraviť sekciu: Externé odkazy" class="mw-editsection-visualeditor"><span>upraviť</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Limita&action=edit&section=5" title="Editovat zdrojový kód sekce Externé odkazy"><span>upraviť zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://matematika.cz/limita-funkce">Limita na matematika.cz</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Zdroj: „<a dir="ltr" href="https://sk.wikipedia.org/w/index.php?title=Limita&oldid=7953653">https://sk.wikipedia.org/w/index.php?title=Limita&oldid=7953653</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%C5%A0peci%C3%A1lne:Kateg%C3%B3rie" title="Špeciálne:Kategórie">Kategórie</a>: <ul><li><a href="/wiki/Kateg%C3%B3ria:Matematick%C3%A1_anal%C3%BDza" title="Kategória:Matematická analýza">Matematická analýza</a></li><li><a href="/wiki/Kateg%C3%B3ria:Vlastnosti_matematick%C3%BDch_funkci%C3%AD" title="Kategória:Vlastnosti matematických funkcií">Vlastnosti matematických funkcií</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Dátum a čas poslednej úpravy tejto stránky: 7. december 2024, 20:09.</li> <li id="footer-info-copyright">Text je dostupný za podmienok <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution/Share-Alike License 4.0 Unported</a>; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke <a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use">Podmienky použitia</a>.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Ochrana osobných údajov</a></li> <li id="footer-places-about"><a href="/wiki/Wikip%C3%A9dia:O">O Wikipédii</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikip%C3%A9dia:Zrieknutie_sa_zodpovednosti">Zrieknutie sa zodpovednosti</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Pravidlá správania</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Vývojári</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/sk.wikipedia.org">Štatistiky</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Vyhlásenie o cookies</a></li> <li id="footer-places-mobileview"><a href="//sk.m.wikipedia.org/w/index.php?title=Limita&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobilné zobrazenie</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-746fb5b8df-9wsrt","wgBackendResponseTime":141,"wgPageParseReport":{"limitreport":{"cputime":"0.033","walltime":"0.113","ppvisitednodes":{"value":205,"limit":1000000},"postexpandincludesize":{"value":0,"limit":2097152},"templateargumentsize":{"value":0,"limit":2097152},"expansiondepth":{"value":2,"limit":100},"expensivefunctioncount":{"value":0,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":1224,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 0.000 1 -total"]},"cachereport":{"origin":"mw-api-ext.codfw.main-65c9bb7d57-p9n85","timestamp":"20241207200924","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Limita","url":"https:\/\/sk.wikipedia.org\/wiki\/Limita","sameAs":"http:\/\/www.wikidata.org\/entity\/Q177239","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q177239","author":{"@type":"Organization","name":"P\u0159isp\u011bvatel\u00e9 projekt\u016f Wikimedia"},"publisher":{"@type":"Organization","name":"nadace Wikimedia","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2006-08-06T16:09:43Z","dateModified":"2024-12-07T20:09:24Z"}</script> </body> </html>