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Gränsvärde – Wikipedia
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<div class="vector-sitenotice-container"> <div id="siteNotice"><!-- CentralNotice --></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="Webbplats"> <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="Innehåll" 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">Innehåll</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">flytta till sidofältet</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">dölj</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">Inledning</div> </a> </li> <li id="toc-Funktioner_av_en_variabel" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Funktioner_av_en_variabel"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Funktioner av en variabel</span> </div> </a> <button aria-controls="toc-Funktioner_av_en_variabel-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>Växla underavsnittet Funktioner av en variabel</span> </button> <ul id="toc-Funktioner_av_en_variabel-sublist" class="vector-toc-list"> <li id="toc-Epsilon-delta-definitionen" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Epsilon-delta-definitionen"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Epsilon-delta-definitionen</span> </div> </a> <ul id="toc-Epsilon-delta-definitionen-sublist" class="vector-toc-list"> <li id="toc-Räkneexempel" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Räkneexempel"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.1</span> <span>Räkneexempel</span> </div> </a> <ul id="toc-Räkneexempel-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Funktioner_av_flera_variabler" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Funktioner_av_flera_variabler"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Funktioner av flera variabler</span> </div> </a> <ul id="toc-Funktioner_av_flera_variabler-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gränsvärden_och_oändligheter" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Gränsvärden_och_oändligheter"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Gränsvärden och oändligheter</span> </div> </a> <button aria-controls="toc-Gränsvärden_och_oändligheter-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>Växla underavsnittet Gränsvärden och oändligheter</span> </button> <ul id="toc-Gränsvärden_och_oändligheter-sublist" class="vector-toc-list"> <li id="toc-Gränsvärden_vid_oändligheten" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gränsvärden_vid_oändligheten"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Gränsvärden vid oändligheten</span> </div> </a> <ul id="toc-Gränsvärden_vid_oändligheten-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Oändliga_gränsvärden" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Oändliga_gränsvärden"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Oändliga gränsvärden</span> </div> </a> <ul id="toc-Oändliga_gränsvärden-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ensidiga_gränsvärden" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ensidiga_gränsvärden"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Ensidiga gränsvärden</span> </div> </a> <button aria-controls="toc-Ensidiga_gränsvärden-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>Växla underavsnittet Ensidiga gränsvärden</span> </button> <ul id="toc-Ensidiga_gränsvärden-sublist" class="vector-toc-list"> <li id="toc-Exempel" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exempel"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Exempel</span> </div> </a> <ul id="toc-Exempel-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Standardgränsvärden" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Standardgränsvärden"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Standardgränsvärden</span> </div> </a> <button aria-controls="toc-Standardgränsvärden-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>Växla underavsnittet Standardgränsvärden</span> </button> <ul id="toc-Standardgränsvärden-sublist" class="vector-toc-list"> <li id="toc-Exempel_på_användning_av_standardgränsvärde" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exempel_på_användning_av_standardgränsvärde"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Exempel på användning av standardgränsvärde</span> </div> </a> <ul id="toc-Exempel_på_användning_av_standardgränsvärde-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Se_även" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Se_även"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Se även</span> </div> </a> <ul id="toc-Se_även-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referenser" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referenser"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Referenser</span> </div> </a> <button aria-controls="toc-Referenser-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>Växla underavsnittet Referenser</span> </button> <ul id="toc-Referenser-sublist" class="vector-toc-list"> <li id="toc-Noter" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Noter"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Noter</span> </div> </a> <ul id="toc-Noter-sublist" class="vector-toc-list"> </ul> </li> </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="Innehåll" 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="Växla innehållsförteckningen" > <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">Växla innehållsförteckningen</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">Gränsvärde</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="Gå till en artikel på ett annat språk. Tillgänglig på 79 språk" > <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 språk</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 – afrikaans" lang="af" hreflang="af" data-title="Limiet" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" 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 – amhariska" lang="am" hreflang="am" data-title="ጥግ" data-language-autonym="አማርኛ" data-language-local-name="amhariska" 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) – arabiska" lang="ar" hreflang="ar" data-title="نهاية (رياضيات)" data-language-autonym="العربية" data-language-local-name="arabiska" 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 – asturiska" lang="ast" hreflang="ast" data-title="Llende matemática" data-language-autonym="Asturianu" data-language-local-name="asturiska" 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) – azerbajdzjanska" lang="az" hreflang="az" data-title="Limit (riyaziyyat)" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajdzjanska" class="interlanguage-link-target"><span>Azərbaycanca</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) – bengali" lang="bn" hreflang="bn" data-title="সীমা (গণিত)" data-language-autonym="বাংলা" data-language-local-name="bengali" class="interlanguage-link-target"><span>বাংলা</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) – basjkiriska" lang="ba" hreflang="ba" data-title="Сикләмә (математика)" data-language-autonym="Башҡортса" data-language-local-name="basjkiriska" 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) – belarusiska" lang="be" hreflang="be" data-title="Ліміт (матэматыка)" data-language-autonym="Беларуская" data-language-local-name="belarusiska" class="interlanguage-link-target"><span>Беларуская</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-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) – bulgariska" lang="bg" hreflang="bg" data-title="Граница (математика)" data-language-autonym="Български" data-language-local-name="bulgariska" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Limes_(matematika)" title="Limes (matematika) – bosniska" lang="bs" hreflang="bs" data-title="Limes (matematika)" data-language-autonym="Bosanski" data-language-local-name="bosniska" 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 – katalanska" lang="ca" hreflang="ca" data-title="Límit" data-language-autonym="Català" data-language-local-name="katalanska" class="interlanguage-link-target"><span>Català</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) – tjuvasjiska" lang="cv" hreflang="cv" data-title="Чикĕ (математика)" data-language-autonym="Чӑвашла" data-language-local-name="tjuvasjiska" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Limita" title="Limita – tjeckiska" lang="cs" hreflang="cs" data-title="Limita" data-language-autonym="Čeština" data-language-local-name="tjeckiska" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Terfan_(mathemateg)" title="Terfan (mathemateg) – walesiska" lang="cy" hreflang="cy" data-title="Terfan (mathemateg)" data-language-autonym="Cymraeg" data-language-local-name="walesiska" 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) – danska" lang="da" hreflang="da" data-title="Grænseværdi (matematik)" data-language-autonym="Dansk" data-language-local-name="danska" class="interlanguage-link-target"><span>Dansk</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) – grekiska" lang="el" hreflang="el" data-title="Όριο (μαθηματικά)" data-language-autonym="Ελληνικά" data-language-local-name="grekiska" 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) – engelska" lang="en" hreflang="en" data-title="Limit (mathematics)" data-language-autonym="English" data-language-local-name="engelska" class="interlanguage-link-target"><span>English</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) – spanska" lang="es" hreflang="es" data-title="Límite (matemática)" data-language-autonym="Español" data-language-local-name="spanska" class="interlanguage-link-target"><span>Español</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-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Limite" title="Limite – baskiska" lang="eu" hreflang="eu" data-title="Limite" data-language-autonym="Euskara" data-language-local-name="baskiska" 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) – persiska" lang="fa" hreflang="fa" data-title="حد (ریاضی)" data-language-autonym="فارسی" data-language-local-name="persiska" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Limite_(math%C3%A9matiques)" title="Limite (mathématiques) – franska" lang="fr" hreflang="fr" data-title="Limite (mathématiques)" data-language-autonym="Français" data-language-local-name="franska" 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) – iriska" lang="ga" hreflang="ga" data-title="Teorainn (matamaitic)" data-language-autonym="Gaeilge" data-language-local-name="iriska" class="interlanguage-link-target"><span>Gaeilge</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 – galiciska" lang="gl" hreflang="gl" data-title="Límite matemático" data-language-autonym="Galego" data-language-local-name="galiciska" class="interlanguage-link-target"><span>Galego</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 – gan" lang="gan" hreflang="gan" data-title="極限" data-language-autonym="贛語" data-language-local-name="gan" class="interlanguage-link-target"><span>贛語</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) – hakka" lang="hak" hreflang="hak" data-title="Khi̍t-han (Sú-ho̍k)" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="hakka" class="interlanguage-link-target"><span>客家語 / Hak-kâ-ngî</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 – koreanska" lang="ko" hreflang="ko" data-title="극한" data-language-autonym="한국어" data-language-local-name="koreanska" class="interlanguage-link-target"><span>한국어</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) – armeniska" lang="hy" hreflang="hy" data-title="Սահման (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="armeniska" class="interlanguage-link-target"><span>Հայերեն</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) – hindi" lang="hi" hreflang="hi" data-title="सीमा (गणित)" data-language-autonym="हिन्दी" data-language-local-name="hindi" 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) – kroatiska" lang="hr" hreflang="hr" data-title="Limes (matematika)" data-language-autonym="Hrvatski" data-language-local-name="kroatiska" class="interlanguage-link-target"><span>Hrvatski</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-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Limit_(matematika)" title="Limit (matematika) – indonesiska" lang="id" hreflang="id" data-title="Limit (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiska" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Markgildi" title="Markgildi – isländska" lang="is" hreflang="is" data-title="Markgildi" data-language-autonym="Íslenska" data-language-local-name="isländska" 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) – italienska" lang="it" hreflang="it" data-title="Limite (matematica)" data-language-autonym="Italiano" data-language-local-name="italienska" class="interlanguage-link-target"><span>Italiano</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) – hebreiska" lang="he" hreflang="he" data-title="גבול (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="hebreiska" class="interlanguage-link-target"><span>עברית</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 – kannada" lang="kn" hreflang="kn" data-title="ಪರಿಮಿತಿ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</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) – georgiska" lang="ka" hreflang="ka" data-title="ზღვარი (მათემატიკა)" data-language-autonym="ქართული" data-language-local-name="georgiska" class="interlanguage-link-target"><span>ქართული</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 – kazakiska" lang="kk" hreflang="kk" data-title="Шек" data-language-autonym="Қазақша" data-language-local-name="kazakiska" class="interlanguage-link-target"><span>Қазақша</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-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Limes_(mathematica)" title="Limes (mathematica) – latin" lang="la" hreflang="la" data-title="Limes (mathematica)" data-language-autonym="Latina" data-language-local-name="latin" class="interlanguage-link-target"><span>Latina</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) – lettiska" lang="lv" hreflang="lv" data-title="Robeža (matemātika)" data-language-autonym="Latviešu" data-language-local-name="lettiska" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Riba_(matematika)" title="Riba (matematika) – litauiska" lang="lt" hreflang="lt" data-title="Riba (matematika)" data-language-autonym="Lietuvių" data-language-local-name="litauiska" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lmo badge-Q17437796 badge-featuredarticle mw-list-item" title="utmärkt artikel"><a href="https://lmo.wikipedia.org/wiki/Limit_(matematega)" title="Limit (matematega) – lombardiska" lang="lmo" hreflang="lmo" data-title="Limit (matematega)" data-language-autonym="Lombard" data-language-local-name="lombardiska" class="interlanguage-link-target"><span>Lombard</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 – ungerska" lang="hu" hreflang="hu" data-title="Határérték" data-language-autonym="Magyar" data-language-local-name="ungerska" class="interlanguage-link-target"><span>Magyar</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 – makedonska" lang="mk" hreflang="mk" data-title="Гранична вредност" data-language-autonym="Македонски" data-language-local-name="makedonska" 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) – marathi" lang="mr" hreflang="mr" data-title="सीमा (गणित)" data-language-autonym="मराठी" data-language-local-name="marathi" 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) – malajiska" lang="ms" hreflang="ms" data-title="Had (matematik)" data-language-autonym="Bahasa Melayu" data-language-local-name="malajiska" 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 – nederländska" lang="nl" hreflang="nl" data-title="Limiet" data-language-autonym="Nederlands" data-language-local-name="nederländska" class="interlanguage-link-target"><span>Nederlands</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 – japanska" lang="ja" hreflang="ja" data-title="極限" data-language-autonym="日本語" data-language-local-name="japanska" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Grenseverdi" title="Grenseverdi – norskt bokmål" lang="nb" hreflang="nb" data-title="Grenseverdi" data-language-autonym="Norsk bokmål" data-language-local-name="norskt bokmål" class="interlanguage-link-target"><span>Norsk bokmål</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 – nynorska" lang="nn" hreflang="nn" data-title="Grense i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="nynorska" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Limit" title="Limit – uzbekiska" lang="uz" hreflang="uz" data-title="Limit" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbekiska" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Limit_(matimatix)" title="Limit (matimatix) – jamaikansk engelsk kreol" lang="jam" hreflang="jam" data-title="Limit (matimatix)" data-language-autonym="Patois" data-language-local-name="jamaikansk engelsk kreol" class="interlanguage-link-target"><span>Patois</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 – kambodjanska" lang="km" hreflang="km" data-title="លីមីត" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="kambodjanska" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Granica_(matematyka)" title="Granica (matematyka) – polska" lang="pl" hreflang="pl" data-title="Granica (matematyka)" data-language-autonym="Polski" data-language-local-name="polska" 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 – portugisiska" lang="pt" hreflang="pt" data-title="Limite" data-language-autonym="Português" data-language-local-name="portugisiska" 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) – rumänska" lang="ro" hreflang="ro" data-title="Limită (matematică)" data-language-autonym="Română" data-language-local-name="rumänska" 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) – ryska" lang="ru" hreflang="ru" data-title="Предел (математика)" data-language-autonym="Русский" data-language-local-name="ryska" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Limiti" title="Limiti – albanska" lang="sq" hreflang="sq" data-title="Limiti" data-language-autonym="Shqip" data-language-local-name="albanska" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/L%C3%ACmiti" title="Lìmiti – sicilianska" lang="scn" hreflang="scn" data-title="Lìmiti" data-language-autonym="Sicilianu" data-language-local-name="sicilianska" class="interlanguage-link-target"><span>Sicilianu</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-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Limita" title="Limita – slovakiska" lang="sk" hreflang="sk" data-title="Limita" data-language-autonym="Slovenčina" data-language-local-name="slovakiska" class="interlanguage-link-target"><span>Slovenčina</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) – centralkurdiska" lang="ckb" hreflang="ckb" data-title="ڕادە (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="centralkurdiska" class="interlanguage-link-target"><span>کوردی</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 – serbiska" lang="sr" hreflang="sr" data-title="Гранична вредност" data-language-autonym="Српски / srpski" data-language-local-name="serbiska" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Limes_(matematika)" title="Limes (matematika) – serbokroatiska" lang="sh" hreflang="sh" data-title="Limes (matematika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroatiska" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Raja-arvo" title="Raja-arvo – finska" lang="fi" hreflang="fi" data-title="Raja-arvo" data-language-autonym="Suomi" data-language-local-name="finska" class="interlanguage-link-target"><span>Suomi</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) – tamil" lang="ta" hreflang="ta" data-title="எல்லை (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Talast_(Tusnakt)" title="Talast (Tusnakt) – kabyliska" lang="kab" hreflang="kab" data-title="Talast (Tusnakt)" data-language-autonym="Taqbaylit" data-language-local-name="kabyliska" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Limit" title="Limit – turkiska" lang="tr" hreflang="tr" data-title="Limit" data-language-autonym="Türkçe" data-language-local-name="turkiska" 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 – ukrainska" lang="uk" hreflang="uk" data-title="Границя" data-language-autonym="Українська" data-language-local-name="ukrainska" 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) – urdu" lang="ur" hreflang="ur" data-title="حد (ریاضی)" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Limite" title="Limite – venetianska" lang="vec" hreflang="vec" data-title="Limite" data-language-autonym="Vèneto" data-language-local-name="venetianska" 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) – vietnamesiska" lang="vi" hreflang="vi" data-title="Giới hạn (toán học)" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamesiska" 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 – västflamländska" lang="vls" hreflang="vls" data-title="Limiet" data-language-autonym="West-Vlams" data-language-local-name="västflamländska" 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） – wu" lang="wuu" hreflang="wuu" data-title="极限（数学）" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</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) – kantonesiska" lang="yue" hreflang="yue" data-title="極限 (數學)" data-language-autonym="粵語" data-language-local-name="kantonesiska" class="interlanguage-link-target"><span>粵語</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-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) – kinesiska" lang="zh" hreflang="zh" data-title="极限 (数学)" data-language-autonym="中文" data-language-local-name="kinesiska" class="interlanguage-link-target"><span>中文</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="Redigera interwikilänkar" class="wbc-editpage">Redigera länkar</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="Namnrymder"> <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/Gr%C3%A4nsv%C3%A4rde" title="Visa innehållssidan [c]" accesskey="c"><span>Artikel</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a 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class="noexcerpt noprint hanvisning_bas"> <dl><dd><i>Den här artikeln handlar om gränsvärden inom matematiken.  För gränsvärden av farliga ämnen, se <a href="/wiki/Gr%C3%A4nsv%C3%A4rde_(arbetsmilj%C3%B6)" title="Gränsvärde (arbetsmiljö)">Gränsvärde (arbetsmiljö)</a>.  </i></dd></dl></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Atan-x.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Atan-x.png/300px-Atan-x.png" decoding="async" width="300" height="179" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Atan-x.png/450px-Atan-x.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Atan-x.png/600px-Atan-x.png 2x" data-file-width="1424" data-file-height="852" /></a><figcaption>arctan(<i>x</i>) är begränsad till ±π/2</figcaption></figure> <p>Ett <b>gränsvärde</b> (<i>limes</i>) (matematisk symbol: lim) för en <a href="/wiki/Funktion" title="Funktion">funktion</a> beskriver funktionens värde när dess <a href="/wiki/Argument_(matematik)" title="Argument (matematik)">argument</a> kommer tillräckligt nära en viss punkt eller växer sig oändligt (eller tillräckligt) stora. Gränsvärden används inom <a href="/wiki/Matematisk_analys" title="Matematisk analys">matematisk analys</a>, bland annat för att definiera <a href="/wiki/Kontinuerlig_funktion" title="Kontinuerlig funktion">kontinuitet</a> och <a href="/wiki/Derivata" title="Derivata">derivata</a>. </p><p>För gränsvärden används notationen </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\rightarrow a}f(x)=A}"> <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>a</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\rightarrow a}f(x)=A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01012d9f7d062f11ff185521f93c6d776d3768fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.099ex; height:3.843ex;" alt="{\displaystyle \lim _{x\rightarrow a}f(x)=A}"></span></dd></dl> <p>alternativt <i>f</i>(<i>x</i>) → <i>A</i> då <i>x</i> → <i>a</i>. </p><p>Båda utläses som ”gränsvärdet av <i>f</i>(<i>x</i>) då <i>x</i> går mot <i>a</i> är lika med <i>A</i>” eller ”limes av <i>f</i>(<i>x</i>) …”, alternativt ”<i>f</i>(<i>x</i>) går mot <i>A</i> då <i>x</i> går mot <i>a</i>”, och innebär att när <i>x</i> är "nästan <i>a</i>" kommer <i>f</i>(<i>x</i>) att vara "nästan <i>A</i>". </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Funktioner_av_en_variabel">Funktioner av en variabel</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&veaction=edit&section=1" title="Redigera avsnitt: Funktioner av en variabel" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&action=edit&section=1" title="Redigera avsnitts källkod: Funktioner av en variabel"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Gr%C3%A4nsv%C3%A4rde-1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Gr%C3%A4nsv%C3%A4rde-1.png/250px-Gr%C3%A4nsv%C3%A4rde-1.png" decoding="async" width="250" height="248" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Gr%C3%A4nsv%C3%A4rde-1.png/375px-Gr%C3%A4nsv%C3%A4rde-1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Gr%C3%A4nsv%C3%A4rde-1.png/500px-Gr%C3%A4nsv%C3%A4rde-1.png 2x" data-file-width="1891" data-file-height="1875" /></a><figcaption>Parametrar för (ε, δ)-definitionen av gränsvärde</figcaption></figure> <p>Antag att <i>f</i> : <b>R</b> → <b>R</b> är definierad på den reella <a href="/wiki/Tallinjen" title="Tallinjen">tallinjen</a> och att <i>a, A</i> ∈ <b>R</b>. Gränsvärdet av <i>f</i>, då <i>x</i> närmar sig <i>a</i>, är <i>A</i> och skrivs </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 a}f(x)=A}"> <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>a</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to a}f(x)=A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65d09d5bfa934b992c44e2ba31287020fdc87b7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.099ex; height:3.843ex;" alt="{\displaystyle \lim _{x\to a}f(x)=A}"></span></dd></dl> <p>om villkoret </p> <dl><dd>För varje reellt <i>ε</i> > 0, existerar ett reellt <i>δ</i> > 0 sådant att för alla reella <i>x</i>, 0 < | <i>x</i> − <i>a</i> | < <i>δ</i> impliceras | <i>f</i>(<i>x</i>) − <i>A</i> | < <i>ε</i></dd></dl> <p>är uppfyllt. Formellt kan villkoret skrivas </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 (\forall \,\varepsilon >0,\,\exists \,\delta >0)\,{\Big (}0<\vert x-a\vert <\delta \,\,\Rightarrow \,\,\vert f(x)-A\vert <\varepsilon {\Big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mspace width="thinmathspace" /> <mi>ε<!-- ε --></mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mspace width="thinmathspace" /> <mi>δ<!-- δ --></mi> <mo>></mo> <mn>0</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mn>0</mn> <mo><</mo> <mo fence="false" stretchy="false">|</mo> <mi>x</mi> <mo>−<!-- − --></mo> <mi>a</mi> <mo fence="false" stretchy="false">|</mo> <mo><</mo> <mi>δ<!-- δ --></mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mo stretchy="false">⇒<!-- ⇒ --></mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">|</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mi>A</mi> <mo fence="false" stretchy="false">|</mo> <mo><</mo> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\forall \,\varepsilon >0,\,\exists \,\delta >0)\,{\Big (}0<\vert x-a\vert <\delta \,\,\Rightarrow \,\,\vert f(x)-A\vert <\varepsilon {\Big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a98a66eb75f92914f6bd249e254ea6d5535b50e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:55.147ex; height:4.843ex;" alt="{\displaystyle (\forall \,\varepsilon >0,\,\exists \,\delta >0)\,{\Big (}0<\vert x-a\vert <\delta \,\,\Rightarrow \,\,\vert f(x)-A\vert <\varepsilon {\Big )}}"></span></dd></dl> <p>Gränsvärdet beror inte av värdet av <i>f</i>(<i>a</i>), eller ens av att <i>a</i> tillhör <i>f</i>:s <a href="/wiki/Definitionsm%C3%A4ngd" title="Definitionsmängd">definitionsmängd</a>. </p><p>Mer generella definitioner är tillämpbara på delmängder av den reella linjen. Låt (<i>a</i>, <i>b</i>) vara ett öppet intervall i <b>R</b> och låt <i>p</i> vara en punkt som tillhör (<i>a</i>, <i>b</i>). Låt <i>f</i> vara en reellvärd funktion definierad på alla (<i>a</i>, <i>b</i>) utom möjligen <i>p</i> själv. Det sägs då att gränsvärdet av <i>f</i>, då <i>x</i> närmar sig <i>p</i>, är <i>A</i> om, för varje reellt <i>ε</i> > <i>0</i>, det existerar ett reellt <i>δ</i> > <i>0</i> sådant att 0 < | <i>x</i> − <i>p</i> | < <i>δ</i> där <i>x</i> ∈ (<i>a</i>, <i>b</i>) implicerar | <i>f</i>(<i>x</i>) − <i>A</i> | < <i>ε</i>. </p><p>Även här beror inte gränsvärdet av att <i>f</i>(<i>p</i>) är väldefinierad. Om till exempel </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^{2}-1}{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> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mrow> <mi>x</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {x^{2}-1}{x-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfb3ccf3f9a48702e27229f4b05ef37f7e88de54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.739ex; height:5.843ex;" alt="{\displaystyle f(x)={\frac {x^{2}-1}{x-1}}}"></span></dd></dl> <p>är <span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;"><i>f</i>(1)</span> odefinierad, men om <span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;"><i>x</i></span> närmar sig 1 tillräckligt mycket, kommer <span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;"><i>f</i>(<i>x</i>)</span> att närma sig 2: </p> <dl><dd><table class="wikitable"> <tbody><tr> <td><i>f</i>(0.9)</td> <td><i>f</i>(0.99)</td> <td><i>f</i>(0.999)</td> <td><i>f</i>(1.0)</td> <td><i>f</i>(1.001)</td> <td><i>f</i>(1.01)</td> <td><i>f</i>(1.1) </td></tr> <tr> <td>1.900</td> <td>1.990</td> <td>1.999</td> <td>odefinierad</td> <td>2.001</td> <td>2.010</td> <td>2.100 </td></tr></tbody></table></dd></dl> <p>En tabell likt den ovan är egentligen inget bevis, men i det här fallet indikerar den sanningen: man kan på ett mer mer rigoröst sätt visa att <span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;"><i>f</i>(<i>x</i>)</span> kan närma sig 2 obegränsat genom att <span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;"><i>x</i></span> obegränsat närmar sig <span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;">1</span>. Med andra ord är </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 1}{\frac {x^{2}-1}{x-1}}=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>1</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mrow> <mi>x</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to 1}{\frac {x^{2}-1}{x-1}}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73aa4e87de950552a3a652f2be98854948366075" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.276ex; height:5.843ex;" alt="{\displaystyle \lim _{x\to 1}{\frac {x^{2}-1}{x-1}}=2}"></span></dd></dl> <p>vilket enkelt inses om täljaren faktoriseras. </p> <div class="mw-heading mw-heading3"><h3 id="Epsilon-delta-definitionen">Epsilon-delta-definitionen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&veaction=edit&section=2" title="Redigera avsnitt: Epsilon-delta-definitionen" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&action=edit&section=2" title="Redigera avsnitts källkod: Epsilon-delta-definitionen"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Augustin-Louis_Cauchy_1901.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Augustin-Louis_Cauchy_1901.jpg/190px-Augustin-Louis_Cauchy_1901.jpg" decoding="async" width="190" height="263" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/d/d3/Augustin-Louis_Cauchy_1901.jpg 1.5x" data-file-width="280" data-file-height="388" /></a><figcaption>Cauchy omkring 1840</figcaption></figure> <p><a href="/wiki/Augustin_Louis_Cauchy" title="Augustin Louis Cauchy">Augustin Louis Cauchy</a>,<sup id="cite_ref-Larson_1-0" class="reference"><a href="#cite_note-Larson-1"><span class="cite-reference-link-bracket">[</span>1<span class="cite-reference-link-bracket">]</span></a></sup> följd av <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a>, formaliserade 1821 definitionen av en funktions gränsvärde, vilken under 1800-talet blev känd som <a href="/wiki/Epsilon-delta-bevis" title="Epsilon-delta-bevis">(ε, δ)-definitionen</a> för gränsvärden. </p><p>Definitionen använder <span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;"><a href="/wiki/%CE%95" class="mw-redirect" title="Ε">ε</a></span> för att representera ett litet positivt tal, så att "<span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;"><i>f</i>(<i>x</i>)</span> kommer godtyckligt nära <span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;"><i>A</i></span>" vilket betyder att <span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;"><i>f</i>(<i>x</i>)</span> eventuellt ligger i intervallet <span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;">(<i>A</i> − ε, <i>A</i> + ε)</span>.<sup id="cite_ref-Larson_1-1" class="reference"><a href="#cite_note-Larson-1"><span class="cite-reference-link-bracket">[</span>1<span class="cite-reference-link-bracket">]</span></a></sup> Frasen ”när <span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;"><i>x</i></span> närmar sig <span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;"><i>c</i></span>" refererar till värden av <span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;"><i>x</i></span> vars avstånd till <span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;"><i>c</i></span> är mindre än ett visst tal <span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;"><a href="/wiki/%CE%94" class="mw-redirect" title="Δ">δ</a></span>: </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 c-\delta <x<c+\delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>−<!-- − --></mo> <mi>δ<!-- δ --></mi> <mo><</mo> <mi>x</mi> <mo><</mo> <mi>c</mi> <mo>+</mo> <mi>δ<!-- δ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c-\delta <x<c+\delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85c6010e4f6b941b94febf565e44cae56a477bc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.318ex; height:2.509ex;" alt="{\displaystyle c-\delta <x<c+\delta }"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Räkneexempel"><span id="R.C3.A4kneexempel"></span>Räkneexempel</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&veaction=edit&section=3" title="Redigera avsnitt: Räkneexempel" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&action=edit&section=3" title="Redigera avsnitts källkod: Räkneexempel"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ett exempel på tillämpning av (ε, δ)-definitionen är ett bevis för att varje linjär funktion </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)=a\,x+b\quad (a,b)\in \mathrm {R} ,a\neq 0}"> <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> <mi>a</mi> <mspace width="thinmathspace" /> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> <mo>,</mo> <mi>a</mi> <mo>≠<!-- ≠ --></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=a\,x+b\quad (a,b)\in \mathrm {R} ,a\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31a02aad988636114d5dea48802f74ccca0ca814" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.77ex; height:2.843ex;" alt="{\displaystyle f(x)=a\,x+b\quad (a,b)\in \mathrm {R} ,a\neq 0}"></span></dd></dl> <p>är kontinuerlig i varje punkt.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-reference-link-bracket">[</span>2<span class="cite-reference-link-bracket">]</span></a></sup> </p><p>Vad som skall visas är att för varje <span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;">ε > 0</span> finns ett <span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;">δ > 0</span> sådant att </p> <dl><dd>när <span style="padding-left:10px;"> </span><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-x_{0}|<\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> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>δ<!-- δ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<|x-x_{0}|<\delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f149a350dfe960048a67f396aba0d1b6b1c581a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.256ex; height:2.843ex;" alt="{\displaystyle 0<|x-x_{0}|<\delta }"></span><span style="padding-left:15px;"> </span> så är <span style="padding-left:15px;"> </span><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)-f(x_{0})|<\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>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ϵ<!-- ϵ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(x)-f(x_{0})|<\epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da00dadadb85424aacbde1b980971f010eb9842b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.066ex; height:2.843ex;" alt="{\displaystyle |f(x)-f(x_{0})|<\epsilon }"></span>.</dd></dl> <p>Vi har </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)-f(x_{0})|=|a\,x+b-(a\,x_{0}+b)|=|a\,x-a\,x_{0}|=|a||x-x_{0}|}"> <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>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mspace width="thinmathspace" /> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mo>−<!-- − --></mo> <mo stretchy="false">(</mo> <mi>a</mi> <mspace width="thinmathspace" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mspace width="thinmathspace" /> <mi>x</mi> <mo>−<!-- − --></mo> <mi>a</mi> <mspace width="thinmathspace" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−<!-- − --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(x)-f(x_{0})|=|a\,x+b-(a\,x_{0}+b)|=|a\,x-a\,x_{0}|=|a||x-x_{0}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/422079a3bd4c72dd9a43e2766f25634ee0613b1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:65.338ex; height:2.843ex;" alt="{\displaystyle |f(x)-f(x_{0})|=|a\,x+b-(a\,x_{0}+b)|=|a\,x-a\,x_{0}|=|a||x-x_{0}|}"></span>.</dd></dl> <p>Det är tydligt att </p> <dl><dd>om <span style="padding-left:10px;"> </span><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-x_{0}|<{\frac {\epsilon }{|a|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−<!-- − --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϵ<!-- ϵ --></mi> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x-x_{0}|<{\frac {\epsilon }{|a|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c8086ff886f4f3c075a945a0e8d5fad4fd09f97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.306ex; height:5.509ex;" alt="{\displaystyle |x-x_{0}|<{\frac {\epsilon }{|a|}}}"></span><span style="padding-left:15px;"> </span> så är <span style="padding-left:15px;"> </span><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)-f(x_{0})|<|a|{\frac {\epsilon }{|a|}}=\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>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϵ<!-- ϵ --></mi> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>ϵ<!-- ϵ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(x)-f(x_{0})|<|a|{\frac {\epsilon }{|a|}}=\epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e26ec0a021e87bff099baf93243f637e8dc27a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.048ex; height:5.509ex;" alt="{\displaystyle |f(x)-f(x_{0})|<|a|{\frac {\epsilon }{|a|}}=\epsilon }"></span>.</dd></dl> <p>Därmed uppfyller <span style="padding-left:10px;"> </span><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 ={\frac {\epsilon }{|a|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ<!-- δ --></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϵ<!-- ϵ --></mi> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta ={\frac {\epsilon }{|a|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1b788f55d8db7382e0ef92461e5645677d8e348" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:7.507ex; height:5.509ex;" alt="{\displaystyle \delta ={\frac {\epsilon }{|a|}}}"></span><span style="padding-left:10px;"> </span> kravet för alla <span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;">ε > 0</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Funktioner_av_flera_variabler">Funktioner av flera variabler</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&veaction=edit&section=4" title="Redigera avsnitt: Funktioner av flera variabler" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&action=edit&section=4" title="Redigera avsnitts källkod: Funktioner av flera variabler"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Para-3D-silver.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Para-3D-silver.png/250px-Para-3D-silver.png" decoding="async" width="250" height="229" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Para-3D-silver.png/375px-Para-3D-silver.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Para-3D-silver.png/500px-Para-3D-silver.png 2x" data-file-width="740" data-file-height="677" /></a><figcaption><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,y)={\frac {x^{2}y}{x^{4}+y^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> </mrow> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)={\frac {x^{2}y}{x^{4}+y^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab2182003fbc04d17a7502e0df9346e0f600445f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.981ex; height:6.343ex;" alt="{\displaystyle f(x,y)={\frac {x^{2}y}{x^{4}+y^{2}}}}"></span></figcaption></figure> <p>Genom att intervallet |<i>x</i> − <i>p</i>| representerar ett avstånd, kan definitionen av gränsvärden för funktioner av en variabel utsträckas till funktioner av flera variabler. </p><p>I fallet med en funktion <i>f</i> : <b>R</b><sup>2</sup> → <b>R</b>, existerar gränsvärdet </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,y)\to (p,q)}f(x,y)=A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{(x,y)\to (p,q)}f(x,y)=A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d68ad96b1ca4b9031e3939169a05c918cafa7ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.293ex; height:4.509ex;" alt="{\displaystyle \lim _{(x,y)\to (p,q)}f(x,y)=A}"></span> <dl><dd>om det för varje <i>ε</i> > 0 existerar ett δ > 0 sådant att för alla</dd> <dd>(<i>x</i>, <i>y</i>) med 0 < ||(<i>x</i>, <i>y</i>) − (<i>p</i>, <i>q</i>)|| < δ, är |<i>f</i>(<i>x</i>, <i>y</i>) − <i>A</i>| < ε</dd></dl></dd> <dd>där ||(<i>x</i>, <i>y</i>) − (<i>p</i>, <i>q</i>)|| representerar det euklidiska avståndet.</dd></dl> <p>Förfarandet kan utökas till godtyckligt antal variabler. </p> <div class="mw-heading mw-heading2"><h2 id="Gränsvärden_och_oändligheter"><span id="Gr.C3.A4nsv.C3.A4rden_och_o.C3.A4ndligheter"></span>Gränsvärden och oändligheter</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&veaction=edit&section=5" title="Redigera avsnitt: Gränsvärden och oändligheter" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&action=edit&section=5" title="Redigera avsnitts källkod: Gränsvärden och oändligheter"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Gränsvärden_vid_oändligheten"><span id="Gr.C3.A4nsv.C3.A4rden_vid_o.C3.A4ndligheten"></span>Gränsvärden vid oändligheten</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&veaction=edit&section=6" title="Redigera avsnitt: Gränsvärden vid oändligheten" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&action=edit&section=6" title="Redigera avsnitts källkod: Gränsvärden vid oändligheten"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Pulse-1x.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Pulse-1x.png/250px-Pulse-1x.png" decoding="async" width="250" height="204" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Pulse-1x.png/375px-Pulse-1x.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Pulse-1x.png/500px-Pulse-1x.png 2x" data-file-width="1208" data-file-height="987" /></a><figcaption>Funktionens gränsvärde existerar vid oändligheten.</figcaption></figure> <p>För den reella funktionen <i>f</i>(<i>x</i>) , betecknas "gränsvärdet av <i>f</i> då <i>x</i> går mot oändligheten är <i>A</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 \lim _{x\to \infty }f(x)=A}"> <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>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to \infty }f(x)=A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3412ba2adbd2a311cfb174bee1a16e45fa2311c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.873ex; height:3.843ex;" alt="{\displaystyle \lim _{x\to \infty }f(x)=A}"></span></dd></dl> <p>vilket betyder att för alla <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<!-- ε --></mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04ec3670b50384a3ce48aca42e7cc5131a06b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}"></span>, existerar ett <i>a</i> sådant att </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)-A|<\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>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε<!-- ε --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(x)-A|<\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fb03afc432e982b0496819e64836d8b04a28547" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.477ex; height:2.843ex;" alt="{\displaystyle |f(x)-A|<\varepsilon }"></span></dd></dl> <p>när <i>x</i> > <i>a</i>. Eller, symboliskt: </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 \forall \varepsilon >0\;\exists a\;\forall x>a:\;|f(x)-A|<\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>ε<!-- ε --></mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mi>a</mi> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo>></mo> <mi>a</mi> <mo>:</mo> <mspace width="thickmathspace" /> <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>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε<!-- ε --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \varepsilon >0\;\exists a\;\forall x>a:\;|f(x)-A|<\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2197a884f0d5da1dfa955d2e44194e8e1b28dda4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.459ex; height:2.843ex;" alt="{\displaystyle \forall \varepsilon >0\;\exists a\;\forall x>a:\;|f(x)-A|<\varepsilon }"></span></dd></dl> <p>På liknande sätt betecknas "gränsvärdet av <i>f</i> då <i>x</i> går mot negativa oändligheten är <i>A</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 \lim _{x\to -\infty }f(x)=A}"> <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>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to -\infty }f(x)=A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59e534d2eda1522b7fb88d8e104ee2f0509301b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.151ex; height:4.009ex;" alt="{\displaystyle \lim _{x\to -\infty }f(x)=A}"></span></dd></dl> <p>vilket betyder att för alla <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<!-- ε --></mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04ec3670b50384a3ce48aca42e7cc5131a06b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}"></span> existerar ett <i>a</i> sådant att <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)-A|<\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>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε<!-- ε --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(x)-A|<\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fb03afc432e982b0496819e64836d8b04a28547" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.477ex; height:2.843ex;" alt="{\displaystyle |f(x)-A|<\varepsilon }"></span> närhelst <i>x</i> < <i>a</i>. Eller i symbolisk form: </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 \forall \varepsilon >0\;\exists a\;\forall x<a:\;|f(x)-A|<\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>ε<!-- ε --></mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mi>a</mi> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo><</mo> <mi>a</mi> <mo>:</mo> <mspace width="thickmathspace" /> <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>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε<!-- ε --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \varepsilon >0\;\exists a\;\forall x<a:\;|f(x)-A|<\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f755588f37cdc901bab1acf5e4db08b3ddf34a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.459ex; height:2.843ex;" alt="{\displaystyle \forall \varepsilon >0\;\exists a\;\forall x<a:\;|f(x)-A|<\varepsilon }"></span></dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:X-inverted.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/X-inverted.png/250px-X-inverted.png" decoding="async" width="250" height="246" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/X-inverted.png/375px-X-inverted.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/X-inverted.png/500px-X-inverted.png 2x" data-file-width="2951" data-file-height="2901" /></a><figcaption>Oändliga gränsvärden för 1/<i>x</i> i en omgivning till 0</figcaption></figure> <p>Exempelvis är </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 }e^{x}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> <mo>−<!-- − --></mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to -\infty }e^{x}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88ca48583c8a957d8e6ec8f4725fa545131f9753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.409ex; height:4.009ex;" alt="{\displaystyle \lim _{x\to -\infty }e^{x}=0}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Oändliga_gränsvärden"><span id="O.C3.A4ndliga_gr.C3.A4nsv.C3.A4rden"></span>Oändliga gränsvärden</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&veaction=edit&section=7" title="Redigera avsnitt: Oändliga gränsvärden" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&action=edit&section=7" title="Redigera avsnitts källkod: Oändliga gränsvärden"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Gränsvärden kan också anta oändliga värden (dessa kallas oftast <i>oegentliga gränsvärden</i>). Till exempel betecknas "gränsvärdet av <i>f</i> då <i>x</i> går mot oändligheten" </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 a}f(x)=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> <mi>a</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to a}f(x)=\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/091e0fe2e72ba5157c3cb080c41f8c677cb7e8c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.68ex; height:3.843ex;" alt="{\displaystyle \lim _{x\to a}f(x)=\infty }"></span></dd></dl> <p>vilket betyder att för alla <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<!-- ε --></mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04ec3670b50384a3ce48aca42e7cc5131a06b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}"></span> existerar ett <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<!-- δ --></mi> <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/595d5cea06fdcaf2642caf549eda2cfc537958a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta >0}"></span> sådant att <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)>\varepsilon }"> <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> <mi>ε<!-- ε --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)>\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff714fea181c420736fe8e9059e74627e8517430" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.6ex; height:2.843ex;" alt="{\displaystyle f(x)>\varepsilon }"></span> när <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x-a|<\delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−<!-- − --></mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>δ<!-- δ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x-a|<\delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6d0084efcea36d9735ea90c17db93e6e5a8cb8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.841ex; height:2.843ex;" alt="{\displaystyle |x-a|<\delta }"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Ensidiga_gränsvärden"><span id="Ensidiga_gr.C3.A4nsv.C3.A4rden"></span>Ensidiga gränsvärden</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&veaction=edit&section=8" title="Redigera avsnitt: Ensidiga gränsvärden" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&action=edit&section=8" title="Redigera avsnitts källkod: Ensidiga gränsvärden"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:SplitGraf.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/SplitGraf.png/250px-SplitGraf.png" decoding="async" width="250" height="177" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/SplitGraf.png/375px-SplitGraf.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/SplitGraf.png/500px-SplitGraf.png 2x" data-file-width="623" data-file-height="440" /></a><figcaption>Gränsvärdena då <i>x</i> → <i>x</i><sub>0</sub><sup>+</sup> respektive <i>x</i> → <i>x</i><sub>0</sub><sup>−</sup> är olika. Därför existerar inte gränsvärdet för <i>x</i> → <i>x</i><sub>0</sub></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Sin-x_x.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Sin-x_x.png/250px-Sin-x_x.png" decoding="async" width="250" height="228" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Sin-x_x.png/375px-Sin-x_x.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Sin-x_x.png/500px-Sin-x_x.png 2x" data-file-width="1916" data-file-height="1747" /></a><figcaption>Funktionen sin 1/<i>x</i> saknar gränsvärde då <i>x</i> → 0<sup>−</sup></figcaption></figure> <p>En funktion kan i en given punkt ha två skilda gränsvärden; ett vänstergränsvärde då <i>x</i> närmar sig punkten ”från vänster” genom ökande värden och ett högergränsvärde då <i>x</i> närmar sig punkten "från höger" genom minskande värden. </p><p>De två gränsvärdena för en reell funktion <i>f</i>(<i>x</i>) av en reell variabel <i>x</i> betecknas med endera av </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 a^{+}}f(x),\quad \lim _{x\downarrow a}\,f(x),\quad \lim _{x\searrow a}\,f(x),\quad \lim _{x{\underset {>}{\to }}a}f(x)}"> <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> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">↓<!-- ↓ --></mo> <mi>a</mi> </mrow> </munder> <mspace width="thinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">↘<!-- ↘ --></mo> <mi>a</mi> </mrow> </munder> <mspace width="thinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo stretchy="false">→<!-- → --></mo> <mo>></mo> </munder> </mrow> <mi>a</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to a^{+}}f(x),\quad \lim _{x\downarrow a}\,f(x),\quad \lim _{x\searrow a}\,f(x),\quad \lim _{x{\underset {>}{\to }}a}f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbe590c4971e08107f7eccd79db88aa3a74933ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:44.853ex; height:5.176ex;" alt="{\displaystyle \lim _{x\to a^{+}}f(x),\quad \lim _{x\downarrow a}\,f(x),\quad \lim _{x\searrow a}\,f(x),\quad \lim _{x{\underset {>}{\to }}a}f(x)}"></span></dd></dl> <p>när <i>x</i> är minskande, eller med endera av </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 a^{-}}f(x),\quad \lim _{x\uparrow a}\,f(x),\quad \lim _{x\nearrow a}\,f(x),\quad \lim _{x{\underset {<}{\to }}a}f(x)}"> <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> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> </mrow> </msup> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">↑<!-- ↑ --></mo> <mi>a</mi> </mrow> </munder> <mspace width="thinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">↗<!-- ↗ --></mo> <mi>a</mi> </mrow> </munder> <mspace width="thinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo stretchy="false">→<!-- → --></mo> <mo><</mo> </munder> </mrow> <mi>a</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to a^{-}}f(x),\quad \lim _{x\uparrow a}\,f(x),\quad \lim _{x\nearrow a}\,f(x),\quad \lim _{x{\underset {<}{\to }}a}f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c29d3009a33137c663445bea5215378662ffe4c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:44.853ex; height:5.176ex;" alt="{\displaystyle \lim _{x\to a^{-}}f(x),\quad \lim _{x\uparrow a}\,f(x),\quad \lim _{x\nearrow a}\,f(x),\quad \lim _{x{\underset {<}{\to }}a}f(x)}"></span></dd></dl> <p>när <i>x</i> är ökande. </p><p>De två ensidiga gränsvärdena existerar och är lika om gränsvärdet till <i>f</i>(<i>x</i>) existerar när <i>x</i> närmar sig <i>a</i>. I vissa fall när gränsvärdet </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 a}f(x)\,}"> <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>a</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to a}f(x)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62dad0b66aeb76b9f20edee37cc09ca1e1c009b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.645ex; height:3.843ex;" alt="{\displaystyle \lim _{x\to a}f(x)\,}"></span></dd></dl> <p>inte existerar, kan höger- och vänstergränsvärden ändå existera. </p><p>Högergränsvärdet kan rigoröst definieras enligt </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 \forall \varepsilon >0\;\exists \delta >0\;\forall x\in I\;(0<x-a<\delta \Rightarrow |f(x)-A|<\varepsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>ε<!-- ε --></mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mi>δ<!-- δ --></mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo>∈<!-- ∈ --></mo> <mi>I</mi> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mn>0</mn> <mo><</mo> <mi>x</mi> <mo>−<!-- − --></mo> <mi>a</mi> <mo><</mo> <mi>δ<!-- δ --></mi> <mo stretchy="false">⇒<!-- ⇒ --></mo> <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>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε<!-- ε --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \varepsilon >0\;\exists \delta >0\;\forall x\in I\;(0<x-a<\delta \Rightarrow |f(x)-A|<\varepsilon )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0896351dfd377e13baf794a91783aaffa16131" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.517ex; height:2.843ex;" alt="{\displaystyle \forall \varepsilon >0\;\exists \delta >0\;\forall x\in I\;(0<x-a<\delta \Rightarrow |f(x)-A|<\varepsilon )}"></span></dd></dl> <p>och vänstergränsvärdet som </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 \forall \varepsilon >0\;\exists \delta >0\;\forall x\in I\;(0<a-x<\delta \Rightarrow |f(x)-A|<\varepsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>ε<!-- ε --></mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mi>δ<!-- δ --></mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo>∈<!-- ∈ --></mo> <mi>I</mi> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mn>0</mn> <mo><</mo> <mi>a</mi> <mo>−<!-- − --></mo> <mi>x</mi> <mo><</mo> <mi>δ<!-- δ --></mi> <mo stretchy="false">⇒<!-- ⇒ --></mo> <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>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε<!-- ε --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \varepsilon >0\;\exists \delta >0\;\forall x\in I\;(0<a-x<\delta \Rightarrow |f(x)-A|<\varepsilon )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/719e3a3271b8b11c808a807cc051d015988d781b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.517ex; height:2.843ex;" alt="{\displaystyle \forall \varepsilon >0\;\exists \delta >0\;\forall x\in I\;(0<a-x<\delta \Rightarrow |f(x)-A|<\varepsilon )}"></span></dd></dl> <p>där <i>I</i> representerar något intervall i <i>f</i>:s <a href="/wiki/Definitionsm%C3%A4ngd" title="Definitionsmängd">definitionsmängd</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Exempel">Exempel</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&veaction=edit&section=9" title="Redigera avsnitt: Exempel" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&action=edit&section=9" title="Redigera avsnitts källkod: Exempel"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Limits-expfunc.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Limits-expfunc.png/250px-Limits-expfunc.png" decoding="async" width="250" height="192" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Limits-expfunc.png/375px-Limits-expfunc.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Limits-expfunc.png/500px-Limits-expfunc.png 2x" data-file-width="1928" data-file-height="1481" /></a><figcaption><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 {\frac {1}{1+2^{-1/x}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{1+2^{-1/x}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/318d56ecd5524bc0693771d9cbfbaa35597d89af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:10.096ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{1+2^{-1/x}}}}"></span></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Sinx_x.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Sinx_x.png/250px-Sinx_x.png" decoding="async" width="250" height="216" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Sinx_x.png/375px-Sinx_x.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Sinx_x.png/500px-Sinx_x.png 2x" data-file-width="957" data-file-height="827" /></a><figcaption><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 {\frac {\sin x}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>x</mi> </mrow> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sin x}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74b7a9205c0aa0b78e764697e518d6268b81e65e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.409ex; height:5.176ex;" alt="{\displaystyle {\frac {\sin x}{x}}}"></span></figcaption></figure> <p>Ett exempel på en funktion som har olika höger- och vänstergränsvärden är </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 0^{+}}{1 \over 1+2^{-1/x}}=1}"> <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> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to 0^{+}}{1 \over 1+2^{-1/x}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa0c539d5b5727e07a5aa20361d11e46c3b791d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.352ex; height:6.009ex;" alt="{\displaystyle \lim _{x\to 0^{+}}{1 \over 1+2^{-1/x}}=1}"></span></dd></dl> <p>medan däremot </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 0^{-}}{1 \over 1+2^{-1/x}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> </mrow> </msup> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to 0^{-}}{1 \over 1+2^{-1/x}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f9cb3b0d43a9b1bd2a7583d7de84f478df5b9b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.352ex; height:6.009ex;" alt="{\displaystyle \lim _{x\to 0^{-}}{1 \over 1+2^{-1/x}}=0}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Standardgränsvärden"><span id="Standardgr.C3.A4nsv.C3.A4rden"></span>Standardgränsvärden</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&veaction=edit&section=10" title="Redigera avsnitt: Standardgränsvärden" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&action=edit&section=10" title="Redigera avsnitts källkod: Standardgränsvärden"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Vissa gränsvärden är särskilt användbara för att bland annat beräkna andra gränsvärden och brukar refereras till som <i>standardgränsvärden</i>, vilka dock inte utgör någon entydigt bestämd grupp. Ett beräkningsuttryck för ett okänt gränsvärde transformeras, om möjligt, så att gränsvärdesdelarna reduceras till ett eller flera standardgränsvärden varefter det sökta gränsvärdet enkelt kan beräknas. En lista över några sådana användbara gränsvärden: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=1}"> <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>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>x</mi> </mrow> <mi>x</mi> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f2fb52f5211c7b7aa69d9e75195afaab5b9d5b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.462ex; height:5.343ex;" alt="{\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=1}"></span></li></ul> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to 0}{\frac {\ln(1+x)}{x}}=1}"> <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>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>x</mi> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to 0}{\frac {\ln(1+x)}{x}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31a32ef3d5fbd5df0406c68a2ad58e415ae641d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.971ex; height:5.843ex;" alt="{\displaystyle \lim _{x\to 0}{\frac {\ln(1+x)}{x}}=1}"></span></li></ul> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to 0}{\frac {e^{x}-1}{x}}=1}"> <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>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mi>x</mi> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to 0}{\frac {e^{x}-1}{x}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38be7d3073e3ee91b5310a8f509dcfc6189916fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.148ex; height:5.343ex;" alt="{\displaystyle \lim _{x\to 0}{\frac {e^{x}-1}{x}}=1}"></span></li></ul> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }{\frac {n^{a}}{b^{n}}}=0\quad b>1}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mi>b</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{\frac {n^{a}}{b^{n}}}=0\quad b>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c943396049438bc47a34066c6a04edf1a166952" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.834ex; height:5.343ex;" alt="{\displaystyle \lim _{n\to \infty }{\frac {n^{a}}{b^{n}}}=0\quad b>1}"></span></li></ul> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }{\frac {a^{n}}{n!}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{\frac {a^{n}}{n!}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/758fa07394f18701c125814eed6b9ea1bc23cc25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.205ex; height:5.343ex;" alt="{\displaystyle \lim _{n\to \infty }{\frac {a^{n}}{n!}}=0}"></span></li></ul> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to \infty }\left(1+{\frac {z}{x}}\right)^{x}=e^{z}}"> <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> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mi>x</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to \infty }\left(1+{\frac {z}{x}}\right)^{x}=e^{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf47ad48e79712a084fd408aba4c04369c2b1618" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.914ex; height:4.843ex;" alt="{\displaystyle \lim _{x\to \infty }\left(1+{\frac {z}{x}}\right)^{x}=e^{z}}"></span></li></ul> <ul><li><a href="/wiki/Stirlings_formel" title="Stirlings formel">Stirlings formel</a>:</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\rightarrow \infty }{n! \over {\sqrt {2\pi n}}\;\left({\frac {n}{e}}\right)^{n}}=1}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π<!-- π --></mi> <mi>n</mi> </msqrt> </mrow> <mspace width="thickmathspace" /> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>e</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\rightarrow \infty }{n! \over {\sqrt {2\pi n}}\;\left({\frac {n}{e}}\right)^{n}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dbb61e09a9eb76ca1dcfd6052ac82c79dec9709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.397ex; height:6.676ex;" alt="{\displaystyle \lim _{n\rightarrow \infty }{n! \over {\sqrt {2\pi n}}\;\left({\frac {n}{e}}\right)^{n}}=1}"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Exempel_på_användning_av_standardgränsvärde"><span id="Exempel_p.C3.A5_anv.C3.A4ndning_av_standardgr.C3.A4nsv.C3.A4rde"></span>Exempel på användning av standardgränsvärde</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&veaction=edit&section=11" title="Redigera avsnitt: Exempel på användning av standardgränsvärde" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&action=edit&section=11" title="Redigera avsnitts källkod: Exempel på användning av standardgränsvärde"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Beräkning av </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 0}{\frac {\sin 5x}{7x}}}"> <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>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡<!-- --></mo> <mn>5</mn> <mi>x</mi> </mrow> <mrow> <mn>7</mn> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to 0}{\frac {\sin 5x}{7x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/751cd2c58ec5235bf3de9f4a00608837293373c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.363ex; height:5.343ex;" alt="{\displaystyle \lim _{x\to 0}{\frac {\sin 5x}{7x}}}"></span></dd></dl> <p>Direkt substitution ger det obestämda uttrycket <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\frac {0}{0}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>0</mn> <mn>0</mn> </mfrac> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\frac {0}{0}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db9e57bd346f546a3354bee7b28a1549b1c0025d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:4.454ex; height:6.176ex;" alt="{\displaystyle \left[{\frac {0}{0}}\right]}"></span>. Gör istället substitutionen </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 t=5x\ \rightarrow x={\cfrac {t}{5}}\quad \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>5</mn> <mi>x</mi> <mtext> </mtext> <mo stretchy="false">→<!-- → --></mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mo stretchy="false">⇒<!-- ⇒ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=5x\ \rightarrow x={\cfrac {t}{5}}\quad \Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df5352ac88b4156c0286e6ad3835e773a9b51850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.343ex; height:7.176ex;" alt="{\displaystyle t=5x\ \rightarrow x={\cfrac {t}{5}}\quad \Rightarrow }"></span></dd> <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 _{t\to 0}{\frac {\sin t}{7\cdot {\cfrac {t}{5}}}}={\frac {5}{7}}\cdot \lim _{t\to 0}{\frac {\sin t}{t}}={\frac {5}{7}}\cdot 1={\frac {5}{7}}}"> <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>t</mi> <mo stretchy="false">→<!-- → --></mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>t</mi> </mrow> <mrow> <mn>7</mn> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>7</mn> </mfrac> </mrow> <mo>⋅<!-- ⋅ --></mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo stretchy="false">→<!-- → --></mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>t</mi> </mrow> <mi>t</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>7</mn> </mfrac> </mrow> <mo>⋅<!-- ⋅ --></mo> <mn>1</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>7</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{t\to 0}{\frac {\sin t}{7\cdot {\cfrac {t}{5}}}}={\frac {5}{7}}\cdot \lim _{t\to 0}{\frac {\sin t}{t}}={\frac {5}{7}}\cdot 1={\frac {5}{7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14c8f4dc2590eba14eff45f16f036cf5ae32666a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:37.64ex; height:10.009ex;" alt="{\displaystyle \lim _{t\to 0}{\frac {\sin t}{7\cdot {\cfrac {t}{5}}}}={\frac {5}{7}}\cdot \lim _{t\to 0}{\frac {\sin t}{t}}={\frac {5}{7}}\cdot 1={\frac {5}{7}}}"></span></dd></dl> <p>där standardgränsvärdet </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 _{t\to 0}{\frac {\sin t}{t}}=1}"> <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>t</mi> <mo stretchy="false">→<!-- → --></mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>t</mi> </mrow> <mi>t</mi> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{t\to 0}{\frac {\sin t}{t}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a1bc35584d1e5ce889f78ed3560c3bd9b253ee8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.796ex; height:5.343ex;" alt="{\displaystyle \lim _{t\to 0}{\frac {\sin t}{t}}=1}"></span></dd></dl> <p>använts. </p> <div class="mw-heading mw-heading2"><h2 id="Se_även"><span id="Se_.C3.A4ven"></span>Se även</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&veaction=edit&section=12" title="Redigera avsnitt: Se även" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&action=edit&section=12" title="Redigera avsnitts källkod: Se även"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/L%27H%C3%B4pitals_regel" title="L'Hôpitals regel">L'Hôpitals regel</a></li> <li><a href="/wiki/Infinitesimalkalkyl" title="Infinitesimalkalkyl">Infinitesimalkalkyl</a></li> <li><a href="/wiki/Inst%C3%A4ngningssatsen" title="Instängningssatsen">Instängningssatsen</a></li> <li><a href="/wiki/Talf%C3%B6ljd" title="Talföljd">Talföljd</a> (konvergens och divergens)</li></ul> <div class="mw-heading mw-heading2"><h2 id="Referenser">Referenser</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&veaction=edit&section=13" title="Redigera avsnitt: Referenser" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&action=edit&section=13" title="Redigera avsnitts källkod: Referenser"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Hylten-Cavallius Sandgren, Matematisk analys, Studentlitteratur 1968</li></ul> <div class="mw-heading mw-heading3"><h3 id="Noter">Noter</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&veaction=edit&section=14" title="Redigera avsnitt: Noter" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Gr%C3%A4nsv%C3%A4rde&action=edit&section=14" title="Redigera avsnitts källkod: Noter"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Larson-1">^ [<a href="#cite_ref-Larson_1-0"><small>a</small></a> <a href="#cite_ref-Larson_1-1"><small>b</small></a>] <span class="reference-text"><cite style="font-style:normal" class="book" id="CITEREFLarsonEdwards2010"><a href="/w/index.php?title=Ron_Larson_(mathematician)&action=edit&redlink=1" class="new" title="Ron Larson (mathematician) [inte skriven än]">Larson, Ron</a>; Edwards, Bruce H. (2010). <i><span>Calculus of a single variable</span></i> (Ninth). <a href="/w/index.php?title=Brooks/Cole&action=edit&redlink=1" class="new" title="Brooks/Cole [inte skriven än]">Brooks/Cole</a>, <a href="/w/index.php?title=Cengage_Learning&action=edit&redlink=1" class="new" title="Cengage Learning [inte skriven än]">Cengage Learning</a>. <a href="/wiki/Special:Bokk%C3%A4llor/978-0-547-20998-2" title="Special:Bokkällor/978-0-547-20998-2">ISBN 978-0-547-20998-2</a></cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus+of+a+single+variable&rft.aulast=Larson&rft.aufirst=Ron&rft.au=Larson%2C+Ron&rft.au=Edwards%2C+Bruce+H.&rft.date=2010&rft.edition=Ninth&rft.pub=%5B%5BBrooks%2FCole%5D%5D%2C+%5B%5BCengage+Learning%5D%5D&rft.isbn=978-0-547-20998-2&rfr_id=info:sid/en.wikipedia.org:Gr%C3%A4nsv%C3%A4rde"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <span class="reference-text">Barile, Margherita. "Epsilon-Delta Proof." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein <a rel="nofollow" class="external free" href="http://mathworld.wolfram.com/Epsilon-DeltaProof.html">http://mathworld.wolfram.com/Epsilon-DeltaProof.html</a></span> </li> </ol></div> <!-- NewPP limit report Parsed by mw‐web.eqiad.main‐5c8dd59f85‐z6cmm Cached time: 20241211134818 Cache expiry: 2592000 Reduced expiry: false Complications: [show‐toc] CPU time usage: 0.100 seconds Real time usage: 0.221 seconds Preprocessor visited node count: 1380/1000000 Post‐expand include size: 8352/2097152 bytes Template argument size: 1558/2097152 bytes Highest expansion depth: 15/100 Expensive parser function count: 0/500 Unstrip recursion depth: 0/20 Unstrip post‐expand size: 4137/5000000 bytes Lua time usage: 0.003/10.000 seconds Lua memory usage: 629779/52428800 bytes Number of Wikibase entities loaded: 0/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 50.757 1 -total 59.81% 30.359 1 Mall:Bokref 56.04% 28.443 1 Mall:Cite_book 50.81% 25.792 1 Mall:Citation/core 14.34% 7.281 1 Mall:Hänvisning 10.87% 5.517 1 Mall:Hänvisning_bas 8.05% 4.087 1 Mall:ISBN 7.72% 3.920 8 Mall:Pad 6.95% 3.526 1 Mall:Hänvisning2 5.86% 2.974 19 Mall:Math --> <!-- Saved in parser cache with key svwiki:pcache:662:|#|:idhash:canonical and timestamp 20241211134818 and revision id 56286143. 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