Limit fungsi - Wikipedia bahasa Indonesia, ensiklopedia bebas

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height: 1.125em;"> <img class="mw-logo-tagline" alt="Ensiklopedia Bebas" src="/static/images/mobile/copyright/wikipedia-tagline-id.svg" width="120" height="14" style="width: 7.5em; height: 0.875em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Istimewa:Pencarian" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Cari di Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Pencarian</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Telusuri Wikipedia" aria-label="Telusuri Wikipedia" autocapitalize="sentences" title="Cari di Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Istimewa:Pencarian"> </div> <button class="cdx-button cdx-search-input__end-button">Cari</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Perkakas pribadi"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Tampilan"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Tampilan" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Tampilan</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=id.wikipedia.org&uselang=id" class=""><span>Menyumbang</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Istimewa:Buat_akun&returnto=Limit+fungsi" title="Anda dianjurkan untuk membuat akun dan masuk log; meskipun, hal itu tidak diwajibkan" class=""><span>Buat akun baru</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Istimewa:Masuk_log&returnto=Limit+fungsi" title="Anda disarankan untuk masuk log, meskipun hal itu tidak diwajibkan. [o]" accesskey="o" class=""><span>Masuk log</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Opsi lainnya" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Perkakas pribadi" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Perkakas pribadi</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Menu pengguna" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=id.wikipedia.org&uselang=id"><span>Menyumbang</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Istimewa:Buat_akun&returnto=Limit+fungsi" title="Anda dianjurkan untuk membuat akun dan masuk log; meskipun, hal itu tidak diwajibkan"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Buat akun baru</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Istimewa:Masuk_log&returnto=Limit+fungsi" title="Anda disarankan untuk masuk log, meskipun hal itu tidak diwajibkan. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Masuk log</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Halaman penyunting yang telah keluar log <a href="/wiki/Bantuan:Pengantar" aria-label="Pelajari lebih lanjut tentang menyunting"><span>pelajari lebih lanjut</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Istimewa:Kontribusi_saya" title="Daftar suntingan yang dibuat dari alamat IP ini [y]" accesskey="y"><span>Kontribusi</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Istimewa:Pembicaraan_saya" title="Pembicaraan tentang suntingan dari alamat IP ini [n]" accesskey="n"><span>Pembicaraan</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </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="Situs"> <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="Daftar isi" 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">Daftar isi</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">sembunyikan</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">Awal</div> </a> </li> <li id="toc-Sejarah" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sejarah"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Sejarah</span> </div> </a> <ul id="toc-Sejarah-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definisi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definisi"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Definisi</span> </div> </a> <button aria-controls="toc-Definisi-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>Gulingkan subbagian Definisi</span> </button> <ul id="toc-Definisi-sublist" class="vector-toc-list"> <li id="toc-Fungsi_pada_garis_bilangan_riil" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fungsi_pada_garis_bilangan_riil"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Fungsi pada garis bilangan riil</span> </div> </a> <ul id="toc-Fungsi_pada_garis_bilangan_riil-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limit_searah" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Limit_searah"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Limit searah</span> </div> </a> <ul id="toc-Limit_searah-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limit_fungsi_pada_ketakhinggaan" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Limit_fungsi_pada_ketakhinggaan"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Limit fungsi pada ketakhinggaan</span> </div> </a> <ul id="toc-Limit_fungsi_pada_ketakhinggaan-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Rumus_biasa" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rumus_biasa"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Rumus biasa</span> </div> </a> <ul id="toc-Rumus_biasa-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rumus" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rumus"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Rumus</span> </div> </a> <ul id="toc-Rumus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lihat_pula" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lihat_pula"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Lihat pula</span> </div> </a> <ul id="toc-Lihat_pula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rujukan" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rujukan"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Rujukan</span> </div> </a> <ul id="toc-Rujukan-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Daftar isi" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Daftar Isi" > <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="Gulingkan daftar isi" > <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">Gulingkan daftar isi</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">Limit fungsi</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="Pergi ke artikel dalam bahasa lain. Terdapat 43 bahasa" > <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-43" 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">43 bahasa</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <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%AF%D8%A7%D9%84%D8%A9" title="نهاية دالة – Arab" lang="ar" hreflang="ar" data-title="نهاية دالة" data-language-autonym="العربية" data-language-local-name="Arab" class="interlanguage-link-target"><span>العربية</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_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%8B%D1%96" title="Ліміт функцыі – Belarusia" lang="be" hreflang="be" data-title="Ліміт функцыі" data-language-autonym="Беларуская" data-language-local-name="Belarusia" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9B%D1%96%D0%BC%D1%96%D1%82_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%8B%D1%96" title="Ліміт функцыі – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Ліміт функцыі" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AB%E0%A6%BE%E0%A6%82%E0%A6%B6%E0%A6%A8%E0%A7%87%E0%A6%B0_%E0%A6%B8%E0%A7%80%E0%A6%AE%E0%A6%BE" title="ফাংশনের সীমা – Bengali" lang="bn" hreflang="bn" data-title="ফাংশনের সীমা" data-language-autonym="বাংলা" data-language-local-name="Bengali" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Grani%C4%8Dna_vrijednost_funkcije" title="Granična vrijednost funkcije – Bosnia" lang="bs" hreflang="bs" data-title="Granična vrijednost funkcije" data-language-autonym="Bosanski" data-language-local-name="Bosnia" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Limita_funkce" title="Limita funkce – Ceko" lang="cs" hreflang="cs" data-title="Limita funkce" data-language-autonym="Čeština" data-language-local-name="Ceko" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8_%D1%87%D0%B8%D0%BA%D0%BA%D0%B8" title="Функци чикки – Chuvash" lang="cv" hreflang="cv" data-title="Функци чикки" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Grenzwert_(Funktion)" title="Grenzwert (Funktion) – Jerman" lang="de" hreflang="de" data-title="Grenzwert (Funktion)" data-language-autonym="Deutsch" data-language-local-name="Jerman" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%8C%CF%81%CE%B9%CE%BF_%CF%83%CF%85%CE%BD%CE%AC%CF%81%CF%84%CE%B7%CF%83%CE%B7%CF%82" title="Όριο συνάρτησης – Yunani" lang="el" hreflang="el" data-title="Όριο συνάρτησης" data-language-autonym="Ελληνικά" data-language-local-name="Yunani" 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_of_a_function" title="Limit of a function – Inggris" lang="en" hreflang="en" data-title="Limit of a function" data-language-autonym="English" data-language-local-name="Inggris" 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_de_una_funci%C3%B3n" title="Límite de una función – Spanyol" lang="es" hreflang="es" data-title="Límite de una función" data-language-autonym="Español" data-language-local-name="Spanyol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Piirv%C3%A4%C3%A4rtus" title="Piirväärtus – Estonia" lang="et" hreflang="et" data-title="Piirväärtus" data-language-autonym="Eesti" data-language-local-name="Estonia" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AD%D8%AF_%D8%AA%D8%A7%D8%A8%D8%B9" title="حد تابع – Persia" lang="fa" hreflang="fa" data-title="حد تابع" data-language-autonym="فارسی" data-language-local-name="Persia" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Funktion_raja-arvo" title="Funktion raja-arvo – Suomi" lang="fi" hreflang="fi" data-title="Funktion raja-arvo" data-language-autonym="Suomi" data-language-local-name="Suomi" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Limite_(math%C3%A9matiques_%C3%A9l%C3%A9mentaires)" title="Limite (mathématiques élémentaires) – Prancis" lang="fr" hreflang="fr" data-title="Limite (mathématiques élémentaires)" data-language-autonym="Français" data-language-local-name="Prancis" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/L%C3%ADmite_dunha_funci%C3%B3n" title="Límite dunha función – Galisia" lang="gl" hreflang="gl" data-title="Límite dunha función" data-language-autonym="Galego" data-language-local-name="Galisia" class="interlanguage-link-target"><span>Galego</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%A9%D7%9C_%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94" title="גבול של פונקציה – Ibrani" lang="he" hreflang="he" data-title="גבול של פונקציה" data-language-autonym="עברית" data-language-local-name="Ibrani" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AB%E0%A4%B2%E0%A4%A8_%E0%A4%95%E0%A5%80_%E0%A4%B8%E0%A5%80%E0%A4%AE%E0%A4%BE" title="फलन की सीमा – Hindi" lang="hi" hreflang="hi" data-title="फलन की सीमा" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Limite_di_una_funzione" title="Limite di una funzione – Italia" lang="it" hreflang="it" data-title="Limite di una funzione" data-language-autonym="Italiano" data-language-local-name="Italia" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%96%A2%E6%95%B0%E3%81%AE%E6%A5%B5%E9%99%90" title="関数の極限 – Jepang" lang="ja" hreflang="ja" data-title="関数の極限" data-language-autonym="日本語" data-language-local-name="Jepang" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A4%E1%83%A3%E1%83%9C%E1%83%A5%E1%83%AA%E1%83%98%E1%83%98%E1%83%A1_%E1%83%96%E1%83%A6%E1%83%95%E1%83%90%E1%83%A0%E1%83%98" title="ფუნქციის ზღვარი – Georgia" lang="ka" hreflang="ka" data-title="ფუნქციის ზღვარი" data-language-autonym="ქართული" data-language-local-name="Georgia" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%95%A8%EC%88%98%EC%9D%98_%EA%B7%B9%ED%95%9C" title="함수의 극한 – Korea" lang="ko" hreflang="ko" data-title="함수의 극한" data-language-autonym="한국어" data-language-local-name="Korea" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%BD%D0%B8%D1%87%D0%BD%D0%B0_%D0%B2%D1%80%D0%B5%D0%B4%D0%BD%D0%BE%D1%81%D1%82_%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%98%D0%B0" title="Гранична вредност на функција – Makedonia" lang="mk" hreflang="mk" data-title="Гранична вредност на функција" data-language-autonym="Македонски" data-language-local-name="Makedonia" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Had_fungsi" title="Had fungsi – Melayu" lang="ms" hreflang="ms" data-title="Had fungsi" data-language-autonym="Bahasa Melayu" data-language-local-name="Melayu" 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#Limiet_van_een_functie" title="Limiet – Belanda" lang="nl" hreflang="nl" data-title="Limiet" data-language-autonym="Nederlands" data-language-local-name="Belanda" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Granica_funkcji" title="Granica funkcji – Polski" lang="pl" hreflang="pl" data-title="Granica funkcji" data-language-autonym="Polski" data-language-local-name="Polski" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Limite_de_uma_fun%C3%A7%C3%A3o" title="Limite de uma função – Portugis" lang="pt" hreflang="pt" data-title="Limite de uma função" data-language-autonym="Português" data-language-local-name="Portugis" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Limit%C4%83_a_unei_func%C8%9Bii" title="Limită a unei funcții – Rumania" lang="ro" hreflang="ro" data-title="Limită a unei funcții" data-language-autonym="Română" data-language-local-name="Rumania" 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_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8" title="Предел функции – Rusia" lang="ru" hreflang="ru" data-title="Предел функции" data-language-autonym="Русский" data-language-local-name="Rusia" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%81%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B7%92%E0%B6%AD%E0%B6%BA%E0%B6%9A_%E0%B7%83%E0%B7%93%E0%B6%B8%E0%B7%8F%E0%B7%80" title="ශ්‍රිතයක සීමාව – Sinhala" lang="si" hreflang="si" data-title="ශ්‍රිතයක සීමාව" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Limit_of_a_function" title="Limit of a function – Simple English" lang="en-simple" hreflang="en-simple" data-title="Limit of a function" 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-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Limita_funkcije" title="Limita funkcije – Slovenia" lang="sl" hreflang="sl" data-title="Limita funkcije" data-language-autonym="Slovenščina" data-language-local-name="Slovenia" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Limiti_i_nj%C3%AB_funksioni" title="Limiti i një funksioni – Albania" lang="sq" hreflang="sq" data-title="Limiti i një funksioni" data-language-autonym="Shqip" data-language-local-name="Albania" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%BD%D0%B8%D1%87%D0%BD%D0%B0_%D0%B2%D1%80%D0%B5%D0%B4%D0%BD%D0%BE%D1%81%D1%82_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%98%D0%B5" title="Гранична вредност функције – Serbia" lang="sr" hreflang="sr" data-title="Гранична вредност функције" data-language-autonym="Српски / srpski" data-language-local-name="Serbia" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AE%BE%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AF%81_%E0%AE%8E%E0%AE%B2%E0%AF%8D%E0%AE%B2%E0%AF%88" title="சார்பு எல்லை – Tamil" lang="ta" hreflang="ta" data-title="சார்பு எல்லை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A5%E0%B8%B4%E0%B8%A1%E0%B8%B4%E0%B8%95%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%9F%E0%B8%B1%E0%B8%87%E0%B8%81%E0%B9%8C%E0%B8%8A%E0%B8%B1%E0%B8%99" title="ลิมิตของฟังก์ชัน – Thai" lang="th" hreflang="th" data-title="ลิมิตของฟังก์ชัน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Hangganan_(kalkulo)" title="Hangganan (kalkulo) – Tagalog" lang="tl" hreflang="tl" data-title="Hangganan (kalkulo)" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Fonksiyonun_limiti" title="Fonksiyonun limiti – Turki" lang="tr" hreflang="tr" data-title="Fonksiyonun limiti" data-language-autonym="Türkçe" data-language-local-name="Turki" 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_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%97_%D0%B2_%D1%82%D0%BE%D1%87%D1%86%D1%96" title="Границя функції в точці – Ukraina" lang="uk" hreflang="uk" data-title="Границя функції в точці" data-language-autonym="Українська" data-language-local-name="Ukraina" 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%AF%D8%A7%D9%84%DB%81_%DA%A9%DB%8C_%D8%AD%D8%AF" title="دالہ کی حد – Urdu" lang="ur" hreflang="ur" data-title="دالہ کی حد" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li 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class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q33540#sitelinks-wikipedia" title="Sunting pranala interwiki" class="wbc-editpage">Sunting pranala</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="Ruang nama"> <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/Limit_fungsi" title="Lihat halaman isi [c]" accesskey="c"><span>Halaman</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Pembicaraan:Limit_fungsi" rel="discussion" title="Pembicaraan halaman isi 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a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse plainlist"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.25em;"><a href="/wiki/Kalkulus" title="Kalkulus">Kalkulus</a></th></tr><tr><td class="sidebar-above" style="padding:0.15em 0.25em 0.3em;font-weight:normal;"> <ul><li><a href="/wiki/Teorema_dasar_kalkulus" title="Teorema dasar kalkulus">Teorema dasar</a></li></ul> <div class="hlist" style="margin-left: 0em;"> <ul><li><a class="mw-selflink selflink">Limit fungsi</a></li> <li><a href="/wiki/Fungsi_kontinu" title="Fungsi kontinu">Kontinuitas</a></li></ul> </div><div class="hlist" style="margin-left: 0em;"> <ul><li><a href="/wiki/Teorema_nilai_purata" title="Teorema nilai purata">Teorema nilai purata</a></li> <li><a href="/wiki/Teorema_Rolle" title="Teorema Rolle">Teorema Rolle</a></li></ul> </div></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;display:block;margin-top:0.65em;"><span style="font-size:110%;"><a href="/wiki/Kalkulus_diferensial" title="Kalkulus diferensial">Diferensial</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r26333518" /><table class="sidebar" style="border-collapse: collapse; border-spacing: 0px; border:none; width:100%; margin:0px; font-size: 100%; clear:none; float:none;"><tbody><tr><th class="sidebar-heading"> Definisi</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Turunan" title="Turunan">Turunan</a> (<a href="/w/index.php?title=Perumuman_dari_turunan&action=edit&redlink=1" class="new" title="Perumuman dari turunan (halaman belum tersedia)">perumuman</a>)</li></ul> <div class="hlist" style="margin-left: 0em; padding:0.1em 0;line-height:1.2em;"> <ul><li><a href="/wiki/Tabel_turunan" title="Tabel turunan">Tabel turunan</a></li> <li><a href="/wiki/Diferensial_(matematika)" class="mw-disambig" title="Diferensial (matematika)">Diferensial</a> <ul><li><a href="/w/index.php?title=Diferensial_(infinitesimal)&action=edit&redlink=1" class="new" title="Diferensial (infinitesimal) (halaman belum tersedia)">infinitesimal</a></li> <li><a href="/wiki/Turunan_fungsi" class="mw-redirect" title="Turunan fungsi">fungsi</a></li> <li><a href="/wiki/Diferensial_total" title="Diferensial total">total</a></li></ul></li></ul> </div></td> </tr><tr><th class="sidebar-heading"> Konsep</th></tr><tr><td class="sidebar-content hlist"> <div class="hlist" style="margin-left: 0em;"> <ul><li><a href="/w/index.php?title=Notasi_untuk_pendiferensialan&action=edit&redlink=1" class="new" title="Notasi untuk pendiferensialan (halaman belum tersedia)">Notasi untuk pendiferensialan</a></li> <li><a href="/wiki/Turunan_kedua" title="Turunan kedua">Turunan kedua</a></li> <li><a href="/wiki/Turunan_ketiga" class="mw-redirect" title="Turunan ketiga">Turunan ketiga</a></li> <li><a href="/w/index.php?title=Perubahan_variabel&action=edit&redlink=1" class="new" title="Perubahan variabel (halaman belum tersedia)">Perubahan variabel</a></li> <li><a href="/wiki/Fungsi_implisit" title="Fungsi implisit">Pendiferensialan implisit</a></li> <li><a href="/w/index.php?title=Laju_yang_berkaitan&action=edit&redlink=1" class="new" title="Laju yang berkaitan (halaman belum tersedia)">Laju yang berkaitan</a></li> <li><a href="/wiki/Teorema_Taylor" title="Teorema Taylor">Teorema Taylor</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Kaidah_diferensiasi" class="mw-redirect" title="Kaidah diferensiasi">Kaidah dan identitas</a></th></tr><tr><td class="sidebar-content hlist"> <div class="hlist" style="margin-left: 0em;"> <ul><li><a href="/w/index.php?title=Kaidah_penjumlahan_dalam_pendiferensialan&action=edit&redlink=1" class="new" title="Kaidah penjumlahan dalam pendiferensialan (halaman belum tersedia)">Kaidah penjumlahan dalam pendiferensialan</a></li> <li><a href="/wiki/Kaidah_darab" title="Kaidah darab">Perkalian</a></li> <li><a href="/wiki/Kaidah_rantai" title="Kaidah rantai">Rantai</a></li> <li><a href="/wiki/Kaidah_pangkat" title="Kaidah pangkat">Pangkat</a></li> <li><a href="/wiki/Kaidah_hasil-bagi" title="Kaidah hasil-bagi">Pembagian</a></li> <li><a href="/w/index.php?title=Rumus_Fa%C3%A0_di_Bruno&action=edit&redlink=1" class="new" title="Rumus Faà di Bruno (halaman belum tersedia)">Rumus Faà di Bruno</a></li></ul> </div></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;"><span style="font-size:110%;"><a href="/wiki/Integral" title="Integral">Integral</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r26333518" /><table class="sidebar" style="border-collapse: collapse; border-spacing: 0px; border:none; width:100%; margin:0px; font-size: 100%; clear:none; float:none;"><tbody><tr><th class="sidebar-heading"> Definisi</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Antiderivatif" class="mw-redirect" title="Antiderivatif">Antiderivatif</a></li> <li><a href="/wiki/Integral" title="Integral">Integral</a> (<a href="/wiki/Integral_takwajar" title="Integral takwajar">takwajar</a>)</li> <li><a href="/wiki/Integral_Riemann" title="Integral Riemann">Integral Riemann</a></li> <li><a href="/wiki/Integrasi_Lebesgue-Stieltjes" title="Integrasi Lebesgue-Stieltjes">Integrasi Lebesgue</a></li> <li><a href="/w/index.php?title=Metode_integrasi_kontur&action=edit&redlink=1" class="new" title="Metode integrasi kontur (halaman belum tersedia)">Integrasi kontur</a></li> <li><a href="/wiki/Tabel_integral" title="Tabel integral">Tabel integral</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Integrasi secara</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Integrasi_parsial" title="Integrasi parsial">parsial</a></li> <li><a href="/wiki/Integrasi_cakram" title="Integrasi cakram">cakram</a></li> <li><a href="/wiki/Integrasi_kulit" title="Integrasi kulit">kulit tabung</a></li> <li><a href="/wiki/Integral_substitusi" title="Integral substitusi">substitusi</a> (<a href="/w/index.php?title=Substitusi_trigonometri&action=edit&redlink=1" class="new" title="Substitusi trigonometri (halaman belum tersedia)">trigonometri</a>)</li> <li><a href="/w/index.php?title=Pecahan_parsial_dalam_integrasi&action=edit&redlink=1" class="new" title="Pecahan parsial dalam integrasi (halaman belum tersedia)">pecahan parsial</a></li> <li><a href="/w/index.php?title=Urutan_integrasi_(kalkulus)&action=edit&redlink=1" class="new" title="Urutan integrasi (kalkulus) (halaman belum tersedia)">Urutan</a></li> <li><a href="/w/index.php?title=Integrasi_dengan_rumus_reduksi&action=edit&redlink=1" class="new" title="Integrasi dengan rumus reduksi (halaman belum tersedia)">Rumus reduksi</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;"><span style="font-size:110%;"><a href="/wiki/Deret_(matematika)" title="Deret (matematika)">Deret</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r26333518" /><table class="sidebar" style="border-collapse: collapse; border-spacing: 0px; border:none; width:100%; margin:0px; font-size: 100%; clear:none; float:none;"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Deret_geometri" class="mw-redirect" title="Deret geometri">geometri</a> (<a href="/w/index.php?title=Deret_aritmetika-geometrik&action=edit&redlink=1" class="new" title="Deret aritmetika-geometrik (halaman belum tersedia)">aritmetika-geometrik</a>)</li> <li><a href="/wiki/Deret_harmonik_(matematika)" title="Deret harmonik (matematika)">harmonik</a></li> <li><a href="/wiki/Deret_selang-seling" title="Deret selang-seling">selang-seling</a></li> <li><a href="/wiki/Deret_pangkat" title="Deret pangkat">pangkat</a></li> <li><a href="/w/index.php?title=Deret_binomial&action=edit&redlink=1" class="new" title="Deret binomial (halaman belum tersedia)">binomial</a></li> <li><a href="/wiki/Deret_Taylor" title="Deret Taylor">Taylor</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Uji_kekonvergenan" title="Uji kekonvergenan">Uji kekonvergenan</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Uji_suku" title="Uji suku">uji suku</a></li> <li><a href="/w/index.php?title=Uji_rasio&action=edit&redlink=1" class="new" title="Uji rasio (halaman belum tersedia)">rasio</a></li> <li><a href="/w/index.php?title=Uji_akar&action=edit&redlink=1" class="new" title="Uji akar (halaman belum tersedia)">akar</a></li> <li><a href="/w/index.php?title=Uji_integral_untuk_kekonvergenan&action=edit&redlink=1" class="new" title="Uji integral untuk kekonvergenan (halaman belum tersedia)">integral</a></li> <li><a href="/w/index.php?title=Uji_perbandingan_langsung&action=edit&redlink=1" class="new" title="Uji perbandingan langsung (halaman belum tersedia)">perbandingan langsung</a></li> <li><br /><a href="/w/index.php?title=Uji_perbandingan_limit&action=edit&redlink=1" class="new" title="Uji perbandingan limit (halaman belum tersedia)">perbandingan limit</a></li> <li><a href="/w/index.php?title=Uji_deret_selang-seling&action=edit&redlink=1" class="new" title="Uji deret selang-seling (halaman belum tersedia)">deret selang-seling</a></li> <li><a href="/w/index.php?title=Uji_kondensasi_Cauchy&action=edit&redlink=1" class="new" title="Uji kondensasi Cauchy (halaman belum tersedia)">kondensasi Cauchy</a></li> <li><a href="/w/index.php?title=Uji_Dirichlet&action=edit&redlink=1" class="new" title="Uji Dirichlet (halaman belum tersedia)">Dirichlet</a></li> <li><a href="/w/index.php?title=Uji_Abel&action=edit&redlink=1" class="new" title="Uji Abel (halaman belum tersedia)">Abel</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;"><span style="font-size:110%;"><a href="/wiki/Kalkulus_vektor" title="Kalkulus vektor">Vektor</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r26333518" /><table class="sidebar" style="border-collapse: collapse; border-spacing: 0px; border:none; width:100%; margin:0px; font-size: 100%; clear:none; float:none;"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Gradien" title="Gradien">Gradien</a></li> <li><a href="/wiki/Divergence" title="Divergence">Divergence</a></li> <li><a href="/w/index.php?title=Keikalan_(matematika)&action=edit&redlink=1" class="new" title="Keikalan (matematika) (halaman belum tersedia)">Keikalan</a></li> <li><a href="/w/index.php?title=Operator_Laplace&action=edit&redlink=1" class="new" title="Operator Laplace (halaman belum tersedia)">Laplace</a></li> <li><a href="/w/index.php?title=Turunan_berarah&action=edit&redlink=1" class="new" title="Turunan berarah (halaman belum tersedia)">berarah</a></li> <li><a href="/w/index.php?title=Identitas_kalkulus_vektor&action=edit&redlink=1" class="new" title="Identitas kalkulus vektor (halaman belum tersedia)">identitas</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Teorema</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/w/index.php?title=Teorema_kedivergenan&action=edit&redlink=1" class="new" title="Teorema kedivergenan (halaman belum tersedia)">Kedivergenan</a></li> <li><a href="/w/index.php?title=Teorema_gradien&action=edit&redlink=1" class="new" title="Teorema gradien (halaman belum tersedia)">Gradien</a></li> <li><a href="/wiki/Teorema_Green" title="Teorema Green">Green</a></li> <li><a href="/wiki/Teorema_Stokes" class="mw-redirect" title="Teorema Stokes">Stokes</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;"><span style="font-size:110%;"><a href="/wiki/Kalkulus_multivariabel" title="Kalkulus multivariabel">Multivariabel</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r26333518" /><table class="sidebar" style="border-collapse: collapse; border-spacing: 0px; border:none; width:100%; margin:0px; font-size: 100%; clear:none; float:none;"><tbody><tr><th class="sidebar-heading"> Formalisme</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Kalkulus_matriks" title="Kalkulus matriks">matriks</a></li> <li><a href="/w/index.php?title=Kalkulus_tensor&action=edit&redlink=1" class="new" title="Kalkulus tensor (halaman belum tersedia)">tensor</a></li> <li><a href="/w/index.php?title=Turunan_luar&action=edit&redlink=1" class="new" title="Turunan luar (halaman belum tersedia)">eksterior</a></li> <li><a href="/w/index.php?title=Kalkulu_geometrik&action=edit&redlink=1" class="new" title="Kalkulu geometrik (halaman belum tersedia)">geometrik</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Definisi</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Turunan_parsial" title="Turunan parsial">Turunan parsial</a></li> <li><a href="/wiki/Integral_lipat" title="Integral lipat">Integral lipat</a></li> <li><a href="/wiki/Integral_garis" title="Integral garis">Integral garis</a></li> <li><a href="/wiki/Permukaan_integral" class="mw-redirect" title="Permukaan integral">Permukaan integral</a></li> <li><a href="/wiki/Integral_volume" title="Integral volume">integral volume</a></li> <li><a href="/wiki/Matriks_dan_determinan_Jacobi" class="mw-redirect" title="Matriks dan determinan Jacobi">Jacobi</a></li> <li><a href="/wiki/Matriks_Hesse" title="Matriks Hesse">Hesse</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;"><span style="font-size:110%;">Khusus</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><div class="hlist" style="margin-left: 0em;"> <ul><li><a href="/w/index.php?title=Kalkulus_fraksional&action=edit&redlink=1" class="new" title="Kalkulus fraksional (halaman belum tersedia)">fraksional</a></li> <li><a href="/w/index.php?title=Kalkulus_Malliavin&action=edit&redlink=1" class="new" title="Kalkulus Malliavin (halaman belum tersedia)">Malliavin</a></li> <li><a href="/w/index.php?title=Kalkulus_stokastik&action=edit&redlink=1" class="new" title="Kalkulus stokastik (halaman belum tersedia)">stokastik</a></li> <li><a href="/w/index.php?title=Kalkulus_variasi&action=edit&redlink=1" class="new" title="Kalkulus variasi (halaman belum tersedia)">variasi</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r18590415">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}.mw-parser-output .infobox .navbar{font-size:100%}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-lihat"><a href="/wiki/Templat:Kalkulus" title="Templat:Kalkulus"><abbr title="Lihat templat ini">l</abbr></a></li><li class="nv-bicara"><a href="/wiki/Pembicaraan_Templat:Kalkulus" title="Pembicaraan Templat:Kalkulus"><abbr title="Diskusikan templat ini">b</abbr></a></li><li class="nv-sunting"><a class="external text" href="https://id.wikipedia.org/w/index.php?title=Templat:Kalkulus&action=edit"><abbr title="Sunting templat ini">s</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Limit</b> suatu <b><a href="/wiki/Fungsi_(matematika)" title="Fungsi (matematika)">fungsi</a></b> merupakan salah satu konsep mendasar dalam <a href="/wiki/Kalkulus" title="Kalkulus">kalkulus</a> dan <a href="/wiki/Analisis_matematika" class="mw-redirect" title="Analisis matematika">analisis</a>, tentang kelakuan suatu fungsi mendekati titik masukan tertentu. </p><p>Suatu fungsi memetakan keluaran <i>f(x)</i> untuk setiap masukan <i>x</i>. Fungsi tersebut memiliki limit <i>L</i> pada titik masukan <i>p</i> bila <i>f(x)</i> "dekat" pada L ketika <i>x</i> dekat pada <i>p</i>. Dengan kata lain, <i>f(x)</i> menjadi semakin dekat kepada <i>L</i> ketika <i>x</i> juga mendekat menuju <i>p</i>. Lebih jauh lagi, bila <i>f</i> diterapkan pada tiap masukan yang <i>cukup</i> dekat pada <i>p</i>, hasilnya adalah keluaran yang (secara sembarang) dekat dengan <i>L</i>. Bila masukan yang <i>dekat</i> pada <i>p</i> ternyata dipetakan pada keluaran yang sangat berbeda, fungsi <i>f</i> dikatakan tidak memiliki limit. </p><p>Definisi limit dirumuskan secara formal mulai <a href="/wiki/Abad_ke-19" class="mw-redirect" title="Abad ke-19">abad ke-19</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Sejarah">Sejarah</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limit_fungsi&veaction=edit&section=1" title="Sunting bagian: Sejarah" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Limit_fungsi&action=edit&section=1" title="Sunting kode sumber bagian: Sejarah"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Meskipun termasuk secara implisit dalam pengembangan kalkulus pada abad ke-17 dan 18, gagasan modern limit fungsi baru dibahas oleh <a href="/wiki/Bernard_Bolzano" title="Bernard Bolzano">Bolzano</a>, yang pada 1817, memperkenalkan dasar-dasar teknik <a href="/w/index.php?title=Epsilon-delta&action=edit&redlink=1" class="new" title="Epsilon-delta (halaman belum tersedia)">epsilon-delta</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Namun karyanya tidak diketahui semasa hidupnya. </p><p><a href="/wiki/Augustin_Louis_Cauchy" title="Augustin Louis Cauchy">Cauchy</a> membahas limit dalam karyanya <i>Cours d'analyse</i> (1821) dan tampaknya telah menyatakan intisari gagasan tersebut, tapi tidak secara sistematis.<sup id="cite_ref-Miller_2-0" class="reference"><a href="#cite_note-Miller-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Presentasi yang ketat terhadap khalayak ramai pertama kali diajukan oleh <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Weirstrass</a> pada dasawarsa 1850-an dan 1860-an,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> dan sejak itu telah menjadi metode baku untuk menerangkan limit. </p><p>Notasi tertulis menggunakan singkatan <b>lim</b> dengan anak panah diperkenalkan oleh <a href="/wiki/G._H._Hardy" class="mw-redirect" title="G. H. Hardy">Hardy</a> dalam bukunya <i>A Course of Pure Mathematics</i> pada tahun 1908.<sup id="cite_ref-Miller_2-1" class="reference"><a href="#cite_note-Miller-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Definisi">Definisi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limit_fungsi&veaction=edit&section=2" title="Sunting bagian: Definisi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Limit_fungsi&action=edit&section=2" title="Sunting kode sumber bagian: Definisi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Berikut beberapa definisi limit fungsi yang umum diterima. </p> <div class="mw-heading mw-heading3"><h3 id="Fungsi_pada_garis_bilangan_riil">Fungsi pada garis <a href="/wiki/Bilangan_riil" title="Bilangan riil">bilangan riil</a></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limit_fungsi&veaction=edit&section=3" title="Sunting bagian: Fungsi pada garis bilangan riil" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Limit_fungsi&action=edit&section=3" title="Sunting kode sumber bagian: Fungsi pada garis bilangan riil"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bila <i>f</i>: <b>R</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e574cc3aa5b4bf5f3f5906caf121a378eef08b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \rightarrow }" /></span> <b>R</b> terdefinisi pada garis <a href="/wiki/Bilangan_riil" title="Bilangan riil">bilangan riil</a>, dan <i>p, L</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \in }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \in }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fe4d5b0a594c1da89b5e78e7dfbeed90bdcc32f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:1.843ex;" alt="{\displaystyle \in }" /></span> <b>R</b> maka kita menyebut <b>limit <i>f</i> ketika <i>x</i> mendekati <i>p</i> adalah <i>L</i></b>, yang ditulis sebagai: </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 p}f(x)=L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>p</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to p}f(x)=L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29dc2f1b0911fbcda671e42b7a9c137c3577daf9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.896ex; height:4.176ex;" alt="{\displaystyle \lim _{x\to p}f(x)=L}" /></span></dd></dl> <p><a href="/wiki/Jika_dan_hanya_jika" title="Jika dan hanya jika">jika dan hanya jika</a> untuk setiap <i>ε</i> > <i>0</i> terdapat <i>δ</i> > <i>0</i> sehingga |<i>x</i> - <i>p</i>|< <i>δ</i> mengimplikasikan bahwa |<i>f</i> (<i>x</i>) - <i>L</i> | < <i>ε</i> . Di sini, baik <i>ε</i> maupun <i>δ</i> merupakan bilangan riil. Perhatikan bahwa nilai limit tidak tergantung pada nilai <i>f</i> (<i>p</i>) </p> <div class="mw-heading mw-heading3"><h3 id="Limit_searah">Limit searah</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limit_fungsi&veaction=edit&section=4" title="Sunting bagian: Limit searah" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Limit_fungsi&action=edit&section=4" title="Sunting kode sumber bagian: Limit searah"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Upper_semi.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/f0/Upper_semi.png" decoding="async" width="214" height="153" class="mw-file-element" data-file-width="214" data-file-height="153" /></a><figcaption>Limit saat: x → x<sub>0</sub><sup>+</sup> ≠ x → x<sub>0</sub><sup>-</sup>. Maka, limit x → x<sub>0</sub> tidak ada.</figcaption></figure> <p>Masukan <i>x</i> dapat mendekati <i>p</i> dari atas (kanan di garis bilangan) atau dari bawah (kiri). Dalam hal ini limit masing-masingnya dapat ditulis sebagai </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 p^{+}}f(x)=L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msup> <mi>p</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> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to p^{+}}f(x)=L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fe0b84549e3caf6ce648960ff5955955c59605c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.098ex; height:4.509ex;" alt="{\displaystyle \lim _{x\to p^{+}}f(x)=L}" /></span></dd></dl> <p>atau </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 p^{-}}f(x)=L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msup> <mi>p</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> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to p^{-}}f(x)=L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ef9ef5a1e71bd645d6d1c7eb2231d54a5debc51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.098ex; height:4.509ex;" alt="{\displaystyle \lim _{x\to p^{-}}f(x)=L}" /></span></dd></dl> <p>Bila kedua limit ini sama nilainya dengan <i>L</i>, maka <i>L</i> dapat diacu sebagai <b>limit <i>f(x)</i> pada <i>p</i> </b>. Sebaliknya, bila keduanya tidak bernilai sama dengan <i>L</i>, maka limit <i>f(x)</i> pada <i>p</i> tidak ada. </p><p>Definisi formal adalah sebagai berikut. Limit <i>f</i>(<i>x</i>) saat <i>x</i> mendekati <i>p</i> dari atas adalah <i>L</i> bila, untuk setiap ε > 0, terdapat sebuah bilangan δ > 0 sedemikian rupa sehingga |<i>f</i>(<i>x</i>) - L| < ε pada saat 0 < <i>x</i> - <i>p</i> < δ. Limit <i>f</i>(<i>x</i>) saat <i>x</i> mendekati <i>p</i> dari bawah adalah <i>L</i> bila, untuk setiap ε > 0, terdapat bilangan δ > 0 sehingga |<i>f</i>(<i>x</i>) - L| < ε bilamana 0 < <i>p</i> - <i>x</i> < δ. </p><p>Bila limitnya tidak ada terdapat osilasi matematis tidak nol. </p> <div class="mw-heading mw-heading3"><h3 id="Limit_fungsi_pada_ketakhinggaan">Limit fungsi pada ketakhinggaan</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limit_fungsi&veaction=edit&section=5" title="Sunting bagian: Limit fungsi pada ketakhinggaan" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Limit_fungsi&action=edit&section=5" title="Sunting kode sumber bagian: Limit fungsi pada ketakhinggaan"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Berkas:Limit-at-infinity-graph.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Limit-at-infinity-graph.png/250px-Limit-at-infinity-graph.png" decoding="async" width="250" height="164" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Limit-at-infinity-graph.png/375px-Limit-at-infinity-graph.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/Limit-at-infinity-graph.png/500px-Limit-at-infinity-graph.png 2x" data-file-width="619" data-file-height="405" /></a><figcaption> Limit fungsi ini ada pada ketakhinggaan.</figcaption></figure> <p>Bila dua unsur, ketakhinggaan positif dan negatif {-∞, +∞}, ditambahkan pada garis bilangan riil, kita dapat mendefinisikan limit fungsi pada ketakhinggaan. Dua unsur tambahan ini bukanlah bilangan, namun berguna dalam memerikan kelakuan limit pada kalkulus dan analisis. </p><p>Bila <i>f</i>(<i>x</i>) adalah fungsi riil, maka <b>limit <i>f</i> saat <i>x</i> mendekati tak hingga</b> adalah <i>L</i>, dilambangkan sebagai: </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)=L,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>L</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to \infty }f(x)=L,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c8a64d74dc879eca107de742a57f3afb88489f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.359ex; height:3.843ex;" alt="{\displaystyle \lim _{x\to \infty }f(x)=L,}" /></span></dd></dl> <p>jika dan hanya jika untuk semua <i>ε</i> > <i>0</i> terdapat <i>S</i> > 0 sedemikian rupa sehingga |<i>f</i> (<i>x</i>) - <i>L</i>| < <i>ε</i> bilamana <i>x > S</i>. </p><p>Dengan cara yang sama, <b>limit <i>f</i> saat <i>x</i> mendekati tak hingga adalah tak hingga</b>, dilambangkan oleh </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)=\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 mathvariant="normal">∞</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to \infty }f(x)=\infty ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08314646580851780b87db428c2f50ad8856e3bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.1ex; height:3.843ex;" alt="{\displaystyle \lim _{x\to \infty }f(x)=\infty ,}" /></span></dd></dl> <p>jika dan hanya jika bila untuk semua <i>R</i> > 0 terdapat <i>S</i> > sedemikian sehingga <i>f</i>(<i>x</i>) > <i>R</i> bilamana <i>x > S</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Rumus_biasa">Rumus biasa</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limit_fungsi&veaction=edit&section=6" title="Sunting bagian: Rumus biasa" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Limit_fungsi&action=edit&section=6" title="Sunting kode sumber bagian: Rumus biasa"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}\lim \limits _{x\to p}&(f(x)+g(x))&=&\lim \limits _{x\to p}f(x)+\lim \limits _{x\to p}g(x)\\\lim \limits _{x\to p}&(f(x)-g(x))&=&\lim \limits _{x\to p}f(x)-\lim \limits _{x\to p}g(x)\\\lim \limits _{x\to p}&(f(x)\cdot g(x))&=&\lim \limits _{x\to p}f(x)\cdot \lim \limits _{x\to p}g(x)\\\lim \limits _{x\to p}&(f(x)/g(x))&=&\lim \limits _{x\to p}f(x)/\lim \limits _{x\to p}g(x)\\\lim \limits _{x\to p}&(f(x))^{n}&=&\lim \limits _{x\to p}f(x))^{n}\\\lim \limits _{x\to p}&{\sqrt[{n}]{(f(x)}}&=&{\sqrt[{n}]{\lim \limits _{x\to p}f(x)}}\\\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>p</mi> </mrow> </munder> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>p</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>p</mi> </mrow> </munder> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>p</mi> </mrow> </munder> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>p</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>p</mi> </mrow> </munder> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>p</mi> </mrow> </munder> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>p</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>p</mi> </mrow> </munder> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>p</mi> </mrow> </munder> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>p</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>p</mi> </mrow> </munder> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>p</mi> </mrow> </munder> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>p</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>p</mi> </mrow> </munder> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>p</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}\lim \limits _{x\to p}&(f(x)+g(x))&=&\lim \limits _{x\to p}f(x)+\lim \limits _{x\to p}g(x)\\\lim \limits _{x\to p}&(f(x)-g(x))&=&\lim \limits _{x\to p}f(x)-\lim \limits _{x\to p}g(x)\\\lim \limits _{x\to p}&(f(x)\cdot g(x))&=&\lim \limits _{x\to p}f(x)\cdot \lim \limits _{x\to p}g(x)\\\lim \limits _{x\to p}&(f(x)/g(x))&=&\lim \limits _{x\to p}f(x)/\lim \limits _{x\to p}g(x)\\\lim \limits _{x\to p}&(f(x))^{n}&=&\lim \limits _{x\to p}f(x))^{n}\\\lim \limits _{x\to p}&{\sqrt[{n}]{(f(x)}}&=&{\sqrt[{n}]{\lim \limits _{x\to p}f(x)}}\\\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80025b440df1a42335642846c9510ce8529bc790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.42ex; margin-bottom: -0.251ex; width:45.368ex; height:28.509ex;" alt="{\displaystyle {\begin{matrix}\lim \limits _{x\to p}&(f(x)+g(x))&=&\lim \limits _{x\to p}f(x)+\lim \limits _{x\to p}g(x)\\\lim \limits _{x\to p}&(f(x)-g(x))&=&\lim \limits _{x\to p}f(x)-\lim \limits _{x\to p}g(x)\\\lim \limits _{x\to p}&(f(x)\cdot g(x))&=&\lim \limits _{x\to p}f(x)\cdot \lim \limits _{x\to p}g(x)\\\lim \limits _{x\to p}&(f(x)/g(x))&=&\lim \limits _{x\to p}f(x)/\lim \limits _{x\to p}g(x)\\\lim \limits _{x\to p}&(f(x))^{n}&=&\lim \limits _{x\to p}f(x))^{n}\\\lim \limits _{x\to p}&{\sqrt[{n}]{(f(x)}}&=&{\sqrt[{n}]{\lim \limits _{x\to p}f(x)}}\\\end{matrix}}}" /></span> </p> <div class="mw-heading mw-heading2"><h2 id="Rumus">Rumus</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limit_fungsi&veaction=edit&section=7" title="Sunting bagian: Rumus" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Limit_fungsi&action=edit&section=7" title="Sunting kode sumber bagian: Rumus"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}\lim \limits _{x\to 0}&{\frac {x}{\sin x}}&=1\\\lim \limits _{x\to 0}&{\frac {x}{\tan x}}&=1\\\lim \limits _{x\to 0}&{\frac {\sin x}{x}}&=1\\\lim \limits _{x\to 0}&{\frac {\tan x}{x}}&=1\\\lim \limits _{x\to \infty }&x\sin({\frac {1}{x}})&=1\\\lim \limits _{x\to \infty }&x\tan({\frac {1}{x}})&=1\\\lim \limits _{x\to 0}&{\frac {ax}{\sin bx}}&={\frac {a}{b}}\\\lim \limits _{x\to 0}&{\frac {ax}{\tan bx}}&={\frac {a}{b}}\\\lim \limits _{x\to 0}&{\frac {\sin ax}{bx}}&={\frac {a}{b}}\\\lim \limits _{x\to 0}&{\frac {\tan ax}{bx}}&={\frac {a}{b}}\\\lim \limits _{x\to \infty }&p^{x}&=0,\qquad -1<p<1\\\lim \limits _{x\to \infty }&{\frac {ax^{m}+b}{px^{n}+q}}&={\frac {a}{p}},\qquad m=n\\\lim \limits _{x\to \infty }&{\sqrt {ax^{2}+bx+c}}-{\sqrt {px^{2}+qx+r}}&={\frac {b-q}{2{\sqrt {a}}}},\qquad a=p\\\lim \limits _{x\to \infty }&{\sqrt[{3}]{ax^{3}+bx^{2}+cx+d}}-{\sqrt[{3}]{px^{3}+qx^{2}+rx+s}}&={\frac {b-q}{3{\sqrt[{3}]{a^{2}}}}},\qquad a=p\\\lim \limits _{x\to \infty }&(1+{\frac {1}{x}})^{x}&=e\\\lim \limits _{x\to 0}&(1+x)^{\frac {1}{x}}&=e\\\lim \limits _{x\to \infty }&(1+{\frac {a}{x}})^{bx}&=e^{ab}\\\lim \limits _{x\to 0}&(1+ax)^{\frac {b}{x}}&=e^{ab}\\\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mi>x</mi> </mfrac> </mrow> </mtd> <mtd> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mi>x</mi> </mfrac> </mrow> </mtd> <mtd> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> </mtd> <mtd> <mi>x</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> </mtd> <mtd> <mi>x</mi> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>x</mi> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>b</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>x</mi> </mrow> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>b</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>a</mi> <mi>x</mi> </mrow> <mrow> <mi>b</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>a</mi> <mi>x</mi> </mrow> <mrow> <mi>b</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> </mtd> <mtd> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mtd> <mtd> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="2em"></mspace> <mo>−</mo> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mi>b</mi> </mrow> <mrow> <mi>p</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mi>q</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>p</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="2em"></mspace> <mi>m</mi> <mo>=</mo> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> </msqrt> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>p</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>q</mi> <mi>x</mi> <mo>+</mo> <mi>r</mi> </msqrt> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>q</mi> </mrow> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="2em"></mspace> <mi>a</mi> <mo>=</mo> <mi>p</mi> </mtd> </mtr> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> <mi>x</mi> <mo>+</mo> <mi>d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mi>p</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>q</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>r</mi> <mi>x</mi> <mo>+</mo> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>q</mi> </mrow> <mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="2em"></mspace> <mi>a</mi> <mo>=</mo> <mi>p</mi> </mtd> </mtr> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mtd> <mtd> <mo>=</mo> <mi>e</mi> </mtd> </mtr> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </msup> </mtd> <mtd> <mo>=</mo> <mi>e</mi> </mtd> </mtr> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>x</mi> </mfrac> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>x</mi> </mrow> </msup> </mtd> <mtd> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>a</mi> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>x</mi> </mfrac> </mrow> </msup> </mtd> <mtd> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}\lim \limits _{x\to 0}&{\frac {x}{\sin x}}&=1\\\lim \limits _{x\to 0}&{\frac {x}{\tan x}}&=1\\\lim \limits _{x\to 0}&{\frac {\sin x}{x}}&=1\\\lim \limits _{x\to 0}&{\frac {\tan x}{x}}&=1\\\lim \limits _{x\to \infty }&x\sin({\frac {1}{x}})&=1\\\lim \limits _{x\to \infty }&x\tan({\frac {1}{x}})&=1\\\lim \limits _{x\to 0}&{\frac {ax}{\sin bx}}&={\frac {a}{b}}\\\lim \limits _{x\to 0}&{\frac {ax}{\tan bx}}&={\frac {a}{b}}\\\lim \limits _{x\to 0}&{\frac {\sin ax}{bx}}&={\frac {a}{b}}\\\lim \limits _{x\to 0}&{\frac {\tan ax}{bx}}&={\frac {a}{b}}\\\lim \limits _{x\to \infty }&p^{x}&=0,\qquad -1<p<1\\\lim \limits _{x\to \infty }&{\frac {ax^{m}+b}{px^{n}+q}}&={\frac {a}{p}},\qquad m=n\\\lim \limits _{x\to \infty }&{\sqrt {ax^{2}+bx+c}}-{\sqrt {px^{2}+qx+r}}&={\frac {b-q}{2{\sqrt {a}}}},\qquad a=p\\\lim \limits _{x\to \infty }&{\sqrt[{3}]{ax^{3}+bx^{2}+cx+d}}-{\sqrt[{3}]{px^{3}+qx^{2}+rx+s}}&={\frac {b-q}{3{\sqrt[{3}]{a^{2}}}}},\qquad a=p\\\lim \limits _{x\to \infty }&(1+{\frac {1}{x}})^{x}&=e\\\lim \limits _{x\to 0}&(1+x)^{\frac {1}{x}}&=e\\\lim \limits _{x\to \infty }&(1+{\frac {a}{x}})^{bx}&=e^{ab}\\\lim \limits _{x\to 0}&(1+ax)^{\frac {b}{x}}&=e^{ab}\\\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf223a044cb88663046007a125f8af74f2b123ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -42.289ex; margin-bottom: -0.216ex; width:75.971ex; height:86.176ex;" alt="{\displaystyle {\begin{matrix}\lim \limits _{x\to 0}&{\frac {x}{\sin x}}&=1\\\lim \limits _{x\to 0}&{\frac {x}{\tan x}}&=1\\\lim \limits _{x\to 0}&{\frac {\sin x}{x}}&=1\\\lim \limits _{x\to 0}&{\frac {\tan x}{x}}&=1\\\lim \limits _{x\to \infty }&x\sin({\frac {1}{x}})&=1\\\lim \limits _{x\to \infty }&x\tan({\frac {1}{x}})&=1\\\lim \limits _{x\to 0}&{\frac {ax}{\sin bx}}&={\frac {a}{b}}\\\lim \limits _{x\to 0}&{\frac {ax}{\tan bx}}&={\frac {a}{b}}\\\lim \limits _{x\to 0}&{\frac {\sin ax}{bx}}&={\frac {a}{b}}\\\lim \limits _{x\to 0}&{\frac {\tan ax}{bx}}&={\frac {a}{b}}\\\lim \limits _{x\to \infty }&p^{x}&=0,\qquad -1<p<1\\\lim \limits _{x\to \infty }&{\frac {ax^{m}+b}{px^{n}+q}}&={\frac {a}{p}},\qquad m=n\\\lim \limits _{x\to \infty }&{\sqrt {ax^{2}+bx+c}}-{\sqrt {px^{2}+qx+r}}&={\frac {b-q}{2{\sqrt {a}}}},\qquad a=p\\\lim \limits _{x\to \infty }&{\sqrt[{3}]{ax^{3}+bx^{2}+cx+d}}-{\sqrt[{3}]{px^{3}+qx^{2}+rx+s}}&={\frac {b-q}{3{\sqrt[{3}]{a^{2}}}}},\qquad a=p\\\lim \limits _{x\to \infty }&(1+{\frac {1}{x}})^{x}&=e\\\lim \limits _{x\to 0}&(1+x)^{\frac {1}{x}}&=e\\\lim \limits _{x\to \infty }&(1+{\frac {a}{x}})^{bx}&=e^{ab}\\\lim \limits _{x\to 0}&(1+ax)^{\frac {b}{x}}&=e^{ab}\\\end{matrix}}}" /></span> </p> <div class="mw-heading mw-heading2"><h2 id="Lihat_pula">Lihat pula</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limit_fungsi&veaction=edit&section=8" title="Sunting bagian: Lihat pula" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Limit_fungsi&action=edit&section=8" title="Sunting kode sumber bagian: Lihat pula"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Aturan_L%27H%C3%B4pital" title="Aturan L'Hôpital">Aturan L'Hôpital</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Rujukan">Rujukan</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limit_fungsi&veaction=edit&section=9" title="Sunting bagian: 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