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data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Indice</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">nascondi</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">Inizio</div> </a> </li> <li id="toc-Storia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Storia"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Storia</span> </div> </a> <ul id="toc-Storia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limite_di_una_successione" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Limite_di_una_successione"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Limite di una successione</span> </div> </a> <ul id="toc-Limite_di_una_successione-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limite_di_una_funzione" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Limite_di_una_funzione"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Limite di una funzione</span> </div> </a> <ul id="toc-Limite_di_una_funzione-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limite_di_un_ultrafiltro" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Limite_di_un_ultrafiltro"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Limite di un ultrafiltro</span> </div> </a> <ul id="toc-Limite_di_un_ultrafiltro-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limite_insiemistico" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Limite_insiemistico"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Limite insiemistico</span> </div> </a> <ul id="toc-Limite_insiemistico-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voci_correlate"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Voci correlate</span> </div> </a> <ul id="toc-Voci_correlate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Altri_progetti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Altri_progetti"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Altri progetti</span> </div> </a> <ul id="toc-Altri_progetti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Collegamenti_esterni" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Collegamenti_esterni"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Collegamenti esterni</span> </div> </a> <ul id="toc-Collegamenti_esterni-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="Indice" 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="Mostra/Nascondi l'indice" > <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">Mostra/Nascondi l'indice</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">Limite (matematica)</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="Vai a una voce in un'altra lingua. Disponibile in 79 lingue" > <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 lingue</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 - amarico" lang="am" hreflang="am" data-title="ጥግ" data-language-autonym="አማርኛ" data-language-local-name="amarico" 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) - arabo" lang="ar" hreflang="ar" data-title="نهاية (رياضيات)" data-language-autonym="العربية" data-language-local-name="arabo" 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 - asturiano" lang="ast" hreflang="ast" data-title="Llende matemática" data-language-autonym="Asturianu" data-language-local-name="asturiano" 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) - azerbaigiano" lang="az" hreflang="az" data-title="Limit (riyaziyyat)" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaigiano" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BA%D0%BB%D3%99%D0%BC%D3%99_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Сикләмә (математика) - baschiro" lang="ba" hreflang="ba" data-title="Сикләмә (математика)" data-language-autonym="Башҡортса" data-language-local-name="baschiro" 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) - bielorusso" lang="be" hreflang="be" data-title="Ліміт (матэматыка)" data-language-autonym="Беларуская" data-language-local-name="bielorusso" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Граница (математика) - bulgaro" lang="bg" hreflang="bg" data-title="Граница (математика)" data-language-autonym="Български" data-language-local-name="bulgaro" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%B8%E0%A5%80%E0%A4%AE%E0%A4%BE_(%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4)" title="सीमा (गणित) - Bhojpuri" lang="bh" hreflang="bh" data-title="सीमा (गणित)" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A7%80%E0%A6%AE%E0%A6%BE_(%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4)" title="সীমা (গণিত) - bengalese" lang="bn" hreflang="bn" data-title="সীমা (গণিত)" data-language-autonym="বাংলা" data-language-local-name="bengalese" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Limes_(matematika)" title="Limes (matematika) - bosniaco" lang="bs" hreflang="bs" data-title="Limes (matematika)" data-language-autonym="Bosanski" data-language-local-name="bosniaco" 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 - catalano" lang="ca" hreflang="ca" data-title="Límit" data-language-autonym="Català" data-language-local-name="catalano" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%95%D8%A7%D8%AF%DB%95_(%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9)" title="ڕادە (ماتماتیک) - curdo centrale" lang="ckb" hreflang="ckb" data-title="ڕادە (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="curdo centrale" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Limita" title="Limita - ceco" lang="cs" hreflang="cs" data-title="Limita" data-language-autonym="Čeština" data-language-local-name="ceco" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A7%D0%B8%D0%BA%C4%95_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Чикĕ (математика) - ciuvascio" lang="cv" hreflang="cv" data-title="Чикĕ (математика)" data-language-autonym="Чӑвашла" data-language-local-name="ciuvascio" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Terfan_(mathemateg)" title="Terfan (mathemateg) - gallese" lang="cy" hreflang="cy" data-title="Terfan (mathemateg)" data-language-autonym="Cymraeg" data-language-local-name="gallese" 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) - danese" lang="da" hreflang="da" data-title="Grænseværdi (matematik)" data-language-autonym="Dansk" data-language-local-name="danese" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Limit" title="Limit - Zazaki" lang="diq" hreflang="diq" data-title="Limit" data-language-autonym="Zazaki" data-language-local-name="Zazaki" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%8C%CF%81%CE%B9%CE%BF_(%CE%BC%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC)" title="Όριο (μαθηματικά) - greco" lang="el" hreflang="el" data-title="Όριο (μαθηματικά)" data-language-autonym="Ελληνικά" data-language-local-name="greco" 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) - inglese" lang="en" hreflang="en" data-title="Limit (mathematics)" data-language-autonym="English" data-language-local-name="inglese" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Limeso" title="Limeso - esperanto" lang="eo" hreflang="eo" data-title="Limeso" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/L%C3%ADmite_(matem%C3%A1tica)" title="Límite (matemática) - spagnolo" lang="es" hreflang="es" data-title="Límite (matemática)" data-language-autonym="Español" data-language-local-name="spagnolo" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Limite" title="Limite - basco" lang="eu" hreflang="eu" data-title="Limite" data-language-autonym="Euskara" data-language-local-name="basco" 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) - persiano" lang="fa" hreflang="fa" data-title="حد (ریاضی)" data-language-autonym="فارسی" data-language-local-name="persiano" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Raja-arvo" title="Raja-arvo - finlandese" lang="fi" hreflang="fi" data-title="Raja-arvo" data-language-autonym="Suomi" data-language-local-name="finlandese" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Limite_(math%C3%A9matiques)" title="Limite (mathématiques) - francese" lang="fr" hreflang="fr" data-title="Limite (mathématiques)" data-language-autonym="Français" data-language-local-name="francese" 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) - irlandese" lang="ga" hreflang="ga" data-title="Teorainn (matamaitic)" data-language-autonym="Gaeilge" data-language-local-name="irlandese" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E6%A5%B5%E9%99%90" title="極限 - 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-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Limit_(mat%C3%A9matik)" title="Limit (matématik) - Guianan Creole" lang="gcr" hreflang="gcr" data-title="Limit (matématik)" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/L%C3%ADmite_matem%C3%A1tico" title="Límite matemático - galiziano" lang="gl" hreflang="gl" data-title="Límite matemático" data-language-autonym="Galego" data-language-local-name="galiziano" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-hak mw-list-item"><a href="https://hak.wikipedia.org/wiki/Khi%CC%8Dt-han_(S%C3%BA-ho%CC%8Dk)" title="Khi̍t-han (Sú-ho̍k) - 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-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) - ebraico" lang="he" hreflang="he" data-title="גבול (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="ebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A5%80%E0%A4%AE%E0%A4%BE_(%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4)" title="सीमा (गणित) - 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) - croato" lang="hr" hreflang="hr" data-title="Limes (matematika)" data-language-autonym="Hrvatski" data-language-local-name="croato" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Hat%C3%A1r%C3%A9rt%C3%A9k" title="Határérték - ungherese" lang="hu" hreflang="hu" data-title="Határérték" data-language-autonym="Magyar" data-language-local-name="ungherese" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8D%D5%A1%D5%B0%D5%B4%D5%A1%D5%B6_(%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Սահման (մաթեմատիկա) - armeno" lang="hy" hreflang="hy" data-title="Սահման (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="armeno" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Limit_(matematika)" title="Limit (matematika) - indonesiano" lang="id" hreflang="id" data-title="Limit (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiano" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Limito" title="Limito - ido" lang="io" hreflang="io" data-title="Limito" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Markgildi" title="Markgildi - islandese" lang="is" hreflang="is" data-title="Markgildi" data-language-autonym="Íslenska" data-language-local-name="islandese" class="interlanguage-link-target"><span>Íslenska</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 - giapponese" lang="ja" hreflang="ja" data-title="極限" data-language-autonym="日本語" data-language-local-name="giapponese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Limit_(matimatix)" title="Limit (matimatix) - creolo giamaicano" lang="jam" hreflang="jam" data-title="Limit (matimatix)" data-language-autonym="Patois" data-language-local-name="creolo giamaicano" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-ka badge-Q70893996 mw-list-item" title=""><a href="https://ka.wikipedia.org/wiki/%E1%83%96%E1%83%A6%E1%83%95%E1%83%90%E1%83%A0%E1%83%98_(%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%90)" title="ზღვარი (მათემატიკა) - georgiano" lang="ka" hreflang="ka" data-title="ზღვარი (მათემატიკა)" data-language-autonym="ქართული" data-language-local-name="georgiano" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Talast_(Tusnakt)" title="Talast (Tusnakt) - cabilo" lang="kab" hreflang="kab" data-title="Talast (Tusnakt)" data-language-autonym="Taqbaylit" data-language-local-name="cabilo" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A8%D0%B5%D0%BA" title="Шек - kazako" lang="kk" hreflang="kk" data-title="Шек" data-language-autonym="Қазақша" data-language-local-name="kazako" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%9B%E1%9E%B8%E1%9E%98%E1%9E%B8%E1%9E%8F" title="លីមីត - khmer" lang="km" hreflang="km" data-title="លីមីត" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AA%E0%B2%B0%E0%B2%BF%E0%B2%AE%E0%B2%BF%E0%B2%A4%E0%B2%BF" title="ಪರಿಮಿತಿ - 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-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B7%B9%ED%95%9C" title="극한 - coreano" lang="ko" hreflang="ko" data-title="극한" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Limes_(mathematica)" title="Limes (mathematica) - latino" lang="la" hreflang="la" data-title="Limes (mathematica)" data-language-autonym="Latina" data-language-local-name="latino" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo badge-Q17437796 badge-featuredarticle mw-list-item" title="voce in vetrina"><a href="https://lmo.wikipedia.org/wiki/Limit_(matematega)" title="Limit (matematega) - lombardo" lang="lmo" hreflang="lmo" data-title="Limit (matematega)" data-language-autonym="Lombard" data-language-local-name="lombardo" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Riba_(matematika)" title="Riba (matematika) - lituano" lang="lt" hreflang="lt" data-title="Riba (matematika)" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Robe%C5%BEa_(matem%C4%81tika)" title="Robeža (matemātika) - lettone" lang="lv" hreflang="lv" data-title="Robeža (matemātika)" data-language-autonym="Latviešu" data-language-local-name="lettone" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%BD%D0%B8%D1%87%D0%BD%D0%B0_%D0%B2%D1%80%D0%B5%D0%B4%D0%BD%D0%BE%D1%81%D1%82" title="Гранична вредност - macedone" lang="mk" hreflang="mk" data-title="Гранична вредност" data-language-autonym="Македонски" data-language-local-name="macedone" 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) - malese" lang="ms" hreflang="ms" data-title="Had (matematik)" data-language-autonym="Bahasa Melayu" data-language-local-name="malese" 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 - olandese" lang="nl" hreflang="nl" data-title="Limiet" data-language-autonym="Nederlands" data-language-local-name="olandese" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Grense_i_matematikk" title="Grense i matematikk - norvegese nynorsk" lang="nn" hreflang="nn" data-title="Grense i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegese nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Grenseverdi" title="Grenseverdi - norvegese bokmål" lang="nb" hreflang="nb" data-title="Grenseverdi" data-language-autonym="Norsk bokmål" data-language-local-name="norvegese bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Granica_(matematyka)" title="Granica (matematyka) - polacco" lang="pl" hreflang="pl" data-title="Granica (matematyka)" data-language-autonym="Polski" data-language-local-name="polacco" 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 - portoghese" lang="pt" hreflang="pt" data-title="Limite" data-language-autonym="Português" data-language-local-name="portoghese" 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) - rumeno" lang="ro" hreflang="ro" data-title="Limită (matematică)" data-language-autonym="Română" data-language-local-name="rumeno" 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) - russo" lang="ru" hreflang="ru" data-title="Предел (математика)" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/L%C3%ACmiti" title="Lìmiti - siciliano" lang="scn" hreflang="scn" data-title="Lìmiti" data-language-autonym="Sicilianu" data-language-local-name="siciliano" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Limes_(matematika)" title="Limes (matematika) - serbo-croato" lang="sh" hreflang="sh" data-title="Limes (matematika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croato" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Limit_(mathematics)" title="Limit (mathematics) - Simple English" lang="en-simple" hreflang="en-simple" data-title="Limit (mathematics)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Limita" title="Limita - slovacco" lang="sk" hreflang="sk" data-title="Limita" data-language-autonym="Slovenčina" data-language-local-name="slovacco" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Limiti" title="Limiti - albanese" lang="sq" hreflang="sq" data-title="Limiti" data-language-autonym="Shqip" data-language-local-name="albanese" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%BD%D0%B8%D1%87%D0%BD%D0%B0_%D0%B2%D1%80%D0%B5%D0%B4%D0%BD%D0%BE%D1%81%D1%82" title="Гранична вредност - serbo" lang="sr" hreflang="sr" data-title="Гранична вредност" data-language-autonym="Српски / srpski" data-language-local-name="serbo" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Gr%C3%A4nsv%C3%A4rde" title="Gränsvärde - svedese" lang="sv" hreflang="sv" data-title="Gränsvärde" data-language-autonym="Svenska" data-language-local-name="svedese" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%8E%E0%AE%B2%E0%AF%8D%E0%AE%B2%E0%AF%88_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="எல்லை (கணிதம்) - 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-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Limit" title="Limit - turco" lang="tr" hreflang="tr" data-title="Limit" data-language-autonym="Türkçe" data-language-local-name="turco" 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 - ucraino" lang="uk" hreflang="uk" data-title="Границя" data-language-autonym="Українська" data-language-local-name="ucraino" 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-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Limit" title="Limit - uzbeco" lang="uz" hreflang="uz" data-title="Limit" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbeco" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Limite" title="Limite - veneto" lang="vec" hreflang="vec" data-title="Limite" data-language-autonym="Vèneto" data-language-local-name="veneto" 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) - vietnamita" lang="vi" hreflang="vi" data-title="Giới hạn (toán học)" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" 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 - fiammingo occidentale" lang="vls" hreflang="vls" data-title="Limiet" data-language-autonym="West-Vlams" data-language-local-name="fiammingo occidentale" 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 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) - cinese" lang="zh" hreflang="zh" data-title="极限 (数学)" data-language-autonym="中文" data-language-local-name="cinese" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%A5%B5%E9%99%90_(%E6%95%B8%E5%AD%B8)" title="極限 (數學) - cantonese" lang="yue" hreflang="yue" data-title="極限 (數學)" data-language-autonym="粵語" data-language-local-name="cantonese" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span 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lang="it" dir="ltr"><p>In <a href="/wiki/Matematica" title="Matematica">matematica</a>, il <b>limite</b> è un concetto basato sull'idea di vicinanza<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> ed è utilizzato per studiare localmente, cioè "vicino" a un punto del <a href="/wiki/Dominio_e_codominio" title="Dominio e codominio">dominio</a>, <a href="/wiki/Funzione_(matematica)" title="Funzione (matematica)">funzioni</a> o, come caso particolare, <a href="/wiki/Successione_(matematica)" title="Successione (matematica)">successioni</a>. </p><p>Il fondamento concettuale del <a href="/wiki/Calcolo_infinitesimale" title="Calcolo infinitesimale">calcolo infinitesimale</a> sta proprio nel concetto di limite e, perciò si può considerare una pietra miliare nella storia del <a href="/wiki/Pensiero_scientifico" class="mw-redirect" title="Pensiero scientifico">pensiero scientifico</a><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>; esso infatti è ricorrente in tutti i rami dell'<a href="/wiki/Analisi_matematica" title="Analisi matematica">analisi matematica</a>, per esempio nel definire la <a href="/wiki/Funzione_continua" title="Funzione continua">continuità</a>, la <a href="/wiki/Derivata" title="Derivata">derivazione</a> e l'<a href="/wiki/Integrale" title="Integrale">integrazione</a>. </p><p>Il concetto di limite di una funzione, più generale del limite di una successione, può essere generalizzato da quello di limite di un <a href="/wiki/Filtro_(matematica)" title="Filtro (matematica)">filtro</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Storia">Storia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limite_(matematica)&veaction=edit&section=1" title="Modifica la sezione Storia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Limite_(matematica)&action=edit&section=1" title="Edit section's source code: Storia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Il concetto di limite era già presente in modo intuitivo nell'antichità, per esempio in <a href="/wiki/Archimede" title="Archimede">Archimede</a> (nel suo <a href="/wiki/Metodo_di_esaustione" title="Metodo di esaustione">metodo di esaustione</a>), e fu utilizzato, anche se non in modo rigoroso, a partire dalla fine del <a href="/wiki/XVII_secolo" title="XVII secolo">XVII secolo</a> da <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a>, <a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Leibniz</a>, <a href="/wiki/Leonhard_Euler" class="mw-redirect" title="Leonhard Euler">Eulero</a> e <a href="/wiki/Jean_Baptiste_Le_Rond_d%27Alembert" title="Jean Baptiste Le Rond d'Alembert">D'Alembert</a>. </p><p>La prima definizione abbastanza rigorosa di limite risale al <a href="/wiki/XIX_secolo" title="XIX secolo">XIX secolo</a> con <a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a>, seguita da una miglior formalizzazione di <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Weierstrass</a>. </p><p>Una completa teoria del limite si ha con <a href="/wiki/Eduard_Heine" title="Eduard Heine">Heine</a>, che nel <a href="/wiki/1872" title="1872">1872</a> pubblicò un lavoro che creò molto interesse all'epoca e nel quale stilò regole e proprietà del limite. Molti altri studiosi si sono interessati al problema del limite, approfondendo l'argomento con lo studio dell'analisi infinitesimale, tra cui <a href="/wiki/Bernard_Bolzano" title="Bernard Bolzano">Bolzano</a>, <a href="/wiki/Julius_Wilhelm_Richard_Dedekind" class="mw-redirect" title="Julius Wilhelm Richard Dedekind">Dedekind</a> e <a href="/wiki/Georg_Cantor" title="Georg Cantor">Cantor</a>. </p><p>Ma solo nel <a href="/wiki/1922" title="1922">1922</a> Eliakim Hastings Moore ed H.L. Smith diedero una nozione generale (<a href="/wiki/Spazio_topologico" title="Spazio topologico">topologica</a>) di limite<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>, ed è quella attualmente utilizzata in matematica. Nel <a href="/wiki/1937" title="1937">1937</a>, <a href="/wiki/Henri_Cartan" title="Henri Cartan">Henri Cartan</a> ne fornì una versione equivalente, basata sul concetto di <a href="/wiki/Filtro_(matematica)" title="Filtro (matematica)">filtro</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Limite_di_una_successione">Limite di una successione</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limite_(matematica)&veaction=edit&section=2" title="Modifica la sezione Limite di una successione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Limite_(matematica)&action=edit&section=2" title="Edit section's source code: Limite di una successione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r130657691">body:not(.skin-minerva) .mw-parser-output .vedi-anche{font-size:95%}</style><style data-mw-deduplicate="TemplateStyles:r139142988">.mw-parser-output .hatnote-content{align-items:center;display:flex}.mw-parser-output .hatnote-icon{flex-shrink:0}.mw-parser-output .hatnote-icon img{display:flex}.mw-parser-output .hatnote-text{font-style:italic}body:not(.skin-minerva) .mw-parser-output .hatnote{border:1px solid #CCC;display:flex;margin:.5em 0;padding:.2em .5em}body:not(.skin-minerva) .mw-parser-output .hatnote-text{padding-left:.5em}body.skin-minerva .mw-parser-output .hatnote-icon{padding-right:8px}body.skin-minerva .mw-parser-output .hatnote-icon img{height:auto;width:16px}body.skin--responsive .mw-parser-output .hatnote a.new{color:#d73333}body.skin--responsive .mw-parser-output .hatnote a.new:visited{color:#a55858}</style> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Limite_di_una_successione" title="Limite di una successione">Limite di una successione</a></b>.</span></div> </div> <p>Una <a href="/wiki/Successione_(matematica)" title="Successione (matematica)">successione</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339d461b497fb8f8670bb29308fe09f0e7bfd34a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.773ex; height:2.843ex;" alt="{\displaystyle \{a_{n}\}}"></span> di <a href="/wiki/Numeri_reali" class="mw-redirect" title="Numeri reali">numeri reali</a> ha come limite il numero <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> se al crescere di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> i termini della successione "sono arbitrariamente vicini" al valore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>. Formalmente, questa nozione è resa chiedendo che per ogni <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04ec3670b50384a3ce48aca42e7cc5131a06b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}"></span> piccolo a piacere esista un <a href="/wiki/Numero_naturale" title="Numero naturale">numero naturale</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> tale che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a_{n}-a|<\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a_{n}-a|<\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b42cb12c091678db79a16d0061d0baaba6c25376" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.994ex; height:2.843ex;" alt="{\displaystyle |a_{n}-a|<\varepsilon }"></span> per ogni <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6592abd10dbd8e25e84efd66c5f4db57d41fe752" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.557ex; height:2.176ex;" alt="{\displaystyle n>N}"></span>. </p><p>Una successione può non avere limite, ad esempio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}=(-1)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}=(-1)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d28e5e8a3c70cf22978a7407aac090da6d914cdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.545ex; height:2.843ex;" alt="{\displaystyle a_{n}=(-1)^{n}}"></span>, data da: </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 1,-1,1,-1,1,-1,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,-1,1,-1,1,-1,\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/711c6b824ff8a01342b93216f0001e8b30b152be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.326ex; height:2.509ex;" alt="{\displaystyle 1,-1,1,-1,1,-1,\ldots }"></span></dd></dl> <p>non ha limite. D'altra parte, se esiste un limite <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>, si dice che la successione <a href="/wiki/Convergenza" title="Convergenza">converge</a> ad <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>; in questo caso, il limite è unico (una successione non può convergere a due valori distinti). Ad esempio, la successione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}=1/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}=1/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df0f29dd89f7151f4246b7ebafb1662a29008b80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.266ex; height:2.843ex;" alt="{\displaystyle a_{n}=1/n}"></span>, data da: </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 1,1/2,1/3,1/4,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,1/2,1/3,1/4,\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d37870c8d78af8447917b2797abd7bfb18527fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.484ex; height:2.843ex;" alt="{\displaystyle 1,1/2,1/3,1/4,\ldots }"></span></dd></dl> <p>converge a zero. </p><p>Considerando uno <a href="/wiki/Spazio_topologico" title="Spazio topologico">spazio topologico</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, una successione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5ea190699149306d242b70439e663559e3ffbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}"></span> con <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }"></span> tende al limite <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6201478b1190a333ea731849684429a697638dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.05ex; height:2.176ex;" alt="{\displaystyle a\in X}"></span> se, comunque si prenda un <a href="/wiki/Intorno" title="Intorno">intorno</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>, esiste un <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> tale per cui <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}\in B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}\in B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7079c82d5be0b8c60b2c3326a1b8ca580e8def66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.153ex; height:2.509ex;" alt="{\displaystyle x_{n}\in B}"></span> per tutti gli <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6592abd10dbd8e25e84efd66c5f4db57d41fe752" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.557ex; height:2.176ex;" alt="{\displaystyle n>N}"></span>, e si scrive: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to +\infty }x_{n}=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>n</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to +\infty }x_{n}=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89f6bc7e78f07f747fe7fb144b8380721f80af9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.815ex; height:3.843ex;" alt="{\displaystyle \lim _{n\to +\infty }x_{n}=a}"></span></dd></dl> <p>Se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> è uno <a href="/wiki/Spazio_di_Hausdorff" title="Spazio di Hausdorff">spazio di Hausdorff</a> il limite di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5ea190699149306d242b70439e663559e3ffbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}"></span> con <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }"></span>, se esiste, è unico. </p> <div class="mw-heading mw-heading2"><h2 id="Limite_di_una_funzione">Limite di una funzione</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limite_(matematica)&veaction=edit&section=3" title="Modifica la sezione Limite di una funzione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Limite_(matematica)&action=edit&section=3" title="Edit section's source code: Limite di una funzione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Limite_di_una_funzione" title="Limite di una funzione">Limite di una funzione</a></b>.</span></div> </div> <p>Il limite di una <a href="/wiki/Funzione_(matematica)" title="Funzione (matematica)">funzione</a> generalizza il limite di una successione di punti in uno <a href="/wiki/Spazio_topologico" title="Spazio topologico">spazio topologico</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>; si considera la successione una funzione <span class="mwe-math-element"><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:\mathbb {N} \to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {N} \to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02c16eeabacc93dc1aff0d78739185121a4246ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.281ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {N} \to Y}"></span> nello spazio topologico <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} \cup \lbrace +\infty \rbrace }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} \cup \lbrace +\infty \rbrace }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/432860abed16ff9b28d84638ed4069989c345eaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.717ex; height:2.843ex;" alt="{\displaystyle \mathbb {N} \cup \lbrace +\infty \rbrace }"></span> con la <a href="/wiki/Topologia_di_sottospazio" title="Topologia di sottospazio">topologia discreta</a>. In tale definizione, un <a href="/wiki/Intorno" title="Intorno">intorno</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }"></span> ha la forma <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lbrace n:n\geq n_{0}\rbrace \cup \lbrace +\infty \rbrace }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>n</mi> <mo>:</mo> <mi>n</mi> <mo>≥</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lbrace n:n\geq n_{0}\rbrace \cup \lbrace +\infty \rbrace }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acbd0478d339bed486f3b75987bc7a46a76f86b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.638ex; height:2.843ex;" alt="{\displaystyle \lbrace n:n\geq n_{0}\rbrace \cup \lbrace +\infty \rbrace }"></span>. </p><p>Siano dati una <a href="/wiki/Funzione_(matematica)" title="Funzione (matematica)">funzione</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\rightarrow \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\rightarrow \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e21f85ac7aa93f713fc356a110fa7a905044215" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.488ex; height:2.509ex;" alt="{\displaystyle f:X\rightarrow \mathbb {R} }"></span> definita su un <a href="/wiki/Sottoinsieme" class="mw-redirect" title="Sottoinsieme">sottoinsieme</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> della <a href="/wiki/Retta_reale" class="mw-redirect" title="Retta reale">retta reale</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> ed un <a href="/wiki/Punto_di_accumulazione" title="Punto di accumulazione">punto di accumulazione</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>. Un <a href="/wiki/Numero_reale" title="Numero reale">numero reale</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> è il limite di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> per <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> tendente a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span> se la distanza fra <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> ed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> è arbitrariamente piccola quando <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> si avvicina a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span>. </p><p>La distanza fra i punti è misurata usando il <a href="/wiki/Valore_assoluto" title="Valore assoluto">valore assoluto</a> della differenza: quindi <span class="mwe-math-element"><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}|}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x-x_{0}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/754a2f965c628a00705dc1c87d2e0bf0c261d850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.848ex; height:2.843ex;" alt="{\displaystyle |x-x_{0}|}"></span> è la distanza fra <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f(x)-l|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(x)-l|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12aed935583d42ac40f0633e52e19ad7af245e98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.245ex; height:2.843ex;" alt="{\displaystyle |f(x)-l|}"></span> è la distanza fra <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> ed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span>. Il concetto di "arbitrariamente piccolo" è espresso formalmente con i <a href="/wiki/Quantificatore" title="Quantificatore">quantificatori</a> "per ogni" (<a href="/wiki/Quantificatore_universale" class="mw-redirect" title="Quantificatore universale">quantificatore universale</a>) ed "esiste" (<a href="/wiki/Quantificatore_esistenziale" class="mw-redirect" title="Quantificatore esistenziale">quantificatore esistenziale</a>). </p><p>Formalmente, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> è limite se per ogni <a href="/wiki/Numero_reale" title="Numero reale">numero reale</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04ec3670b50384a3ce48aca42e7cc5131a06b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}"></span> piccolo a piacere esiste un altro numero reale positivo <span class="mwe-math-element"><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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"></span> tale che: </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)-l|<\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(x)-l|<\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/041a299811036369baf472fb106deea21255ecb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.427ex; height:2.843ex;" alt="{\displaystyle |f(x)-l|<\varepsilon }"></span> per ogni <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> con <span class="mwe-math-element"><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</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>.</dd></dl> <p>In questo caso si scrive: </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 x_{0}}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> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </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 x_{0}}f(x)=l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7d064249e22c27f576b107c2118a40bfc068095" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.952ex; height:4.176ex;" alt="{\displaystyle \lim _{x\to x_{0}}f(x)=l}"></span></dd></dl> <p>La definizione di limite di una funzione è comoda per formalizzare il concetto di <a href="/wiki/Funzione_continua" title="Funzione continua">funzione continua</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Limite_di_un_ultrafiltro">Limite di un ultrafiltro</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limite_(matematica)&veaction=edit&section=4" title="Modifica la sezione Limite di un ultrafiltro" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Limite_(matematica)&action=edit&section=4" title="Edit section's source code: Limite di un ultrafiltro"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dato uno <a href="/wiki/Spazio_topologico" title="Spazio topologico">spazio topologico</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/627ce34fb2db641fafd1e88cf307a668d1789ad4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.46ex; height:2.843ex;" alt="{\displaystyle (X,T)}"></span>, un punto <span class="mwe-math-element"><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\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}"></span> è il limite di un <a href="/wiki/Ultrafiltro" title="Ultrafiltro">ultrafiltro</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> su <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> se ogni <a href="/wiki/Intorno" title="Intorno">intorno</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> appartiene a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span>. </p><p>Il limite di una funzione rispetto ad un <a href="/wiki/Filtro_(matematica)" title="Filtro (matematica)">filtro</a> è definito considerando una funzione <span class="mwe-math-element"><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:B\subset X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>B</mi> <mo>⊂</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:B\subset X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5f8a199f6a6162e5da88e9775aab8173a7251fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.446ex; height:2.509ex;" alt="{\displaystyle f:B\subset X\to Y}"></span> tra spazi topologici e un filtro <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> su <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>. Il punto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cee1c0ec36a82f33f5e3d7434d5667881b4ec323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.769ex; height:2.509ex;" alt="{\displaystyle y\in Y}"></span> è il limite di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> in <span class="mwe-math-element"><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\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}"></span> rispetto ad <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> è il limite di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> è il limite di <span class="mwe-math-element"><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(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(U)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/562e763c2291125cfb06f14882bc9f9aba8a7d7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.87ex; height:2.843ex;" alt="{\displaystyle f(U)}"></span>. Si scrive in tal caso: </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 _{U}f(x)=y}"> <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>U</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{U}f(x)=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2283d27f33a93721d2bbc8b568795079ae8a22d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.288ex; height:4.009ex;" alt="{\displaystyle \lim _{U}f(x)=y}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Limite_insiemistico">Limite insiemistico</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limite_(matematica)&veaction=edit&section=5" title="Modifica la sezione Limite insiemistico" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Limite_(matematica)&action=edit&section=5" title="Edit section's source code: Limite insiemistico"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Limite_insiemistico" title="Limite insiemistico">Limite insiemistico</a></b>.</span></div> </div> <p>Il concetto di limite si estende anche alle successioni di <a href="/wiki/Insieme" title="Insieme">insiemi</a> attraverso le nozioni di <a href="/wiki/Limite_superiore_e_limite_inferiore" title="Limite superiore e limite inferiore">limite superiore e limite inferiore</a>: data una successione di insiemi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{A_{n}\}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{A_{n}\}_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a62af4f5287dc020b3ff5dcbb74e08e29569d680" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.505ex; height:2.843ex;" alt="{\displaystyle \{A_{n}\}_{n}}"></span>, l'insieme limite è definito come l'insieme che intuitivamente contiene gli elementi che stanno nel maggior numero di insiemi della successione. Formalmente, una successione di insiemi si dice possedere limite se vale la seguente uguaglianza: </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 {\bigcap _{n=1}^{\infty }}\left({\bigcup _{m=n}^{\infty }}A_{m}\right)={\bigcup _{n=1}^{\infty }}\left({\bigcap _{m=n}^{\infty }}A_{m}\right)=:\lim _{n\to \infty }A_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> </mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> </mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=:</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigcap _{n=1}^{\infty }}\left({\bigcup _{m=n}^{\infty }}A_{m}\right)={\bigcup _{n=1}^{\infty }}\left({\bigcap _{m=n}^{\infty }}A_{m}\right)=:\lim _{n\to \infty }A_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/527c5558b656118e1a12fbf3b38ef400c97e7d72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:43.027ex; height:7.509ex;" alt="{\displaystyle {\bigcap _{n=1}^{\infty }}\left({\bigcup _{m=n}^{\infty }}A_{m}\right)={\bigcup _{n=1}^{\infty }}\left({\bigcap _{m=n}^{\infty }}A_{m}\right)=:\lim _{n\to \infty }A_{n}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limite_(matematica)&veaction=edit&section=6" title="Modifica la sezione Note" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Limite_(matematica)&action=edit&section=6" title="Edit section's source code: Note"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1"><b>^</b></a> <span class="reference-text"><cite class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://www.britannica.com/science/limit-mathematics"><span style="font-style:italic;">Limit | Definition, Example, & Facts | Britannica</span></a>, su <span style="font-style:italic;">www.britannica.com</span>, 6 agosto 2024. <small>URL consultato il 2 settembre 2024</small>.</cite></span> </li> <li id="cite_note-2"><a href="#cite_ref-2"><b>^</b></a> <span class="reference-text"><cite class="citation libro" style="font-style:normal"> Marco Bramanti, Carlo D. Pagani e Sandro Salsa, <span style="font-style:italic;">Analisi matematica 1</span>, Prima edizione, <a href="/wiki/Bologna" title="Bologna">Bologna</a>-<a href="/wiki/Milano" title="Milano">Milano</a>, <a href="/wiki/Zanichelli" title="Zanichelli">Zanichelli</a>, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-88-08-06485-1" title="Speciale:RicercaISBN/978-88-08-06485-1">978-88-08-06485-1</a>.</cite></span> </li> <li id="cite_note-3"><a href="#cite_ref-3"><b>^</b></a> <span class="reference-text">Si veda Moore, Smith <i>A General Theory of Limits</i></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limite_(matematica)&veaction=edit&section=7" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Limite_(matematica)&action=edit&section=7" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation libro" style="font-style:normal"> Carlo Domenico Pagani e <a href="/wiki/Sandro_Salsa" title="Sandro Salsa">Sandro Salsa</a>, <span style="font-style:italic;">Analisi matematica 1</span>, Seconda edizione, <a href="/wiki/Bologna" title="Bologna">Bologna</a>, <a href="/wiki/Zanichelli" title="Zanichelli">Zanichelli</a>, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-88-08-15133-9" title="Speciale:RicercaISBN/978-88-08-15133-9">978-88-08-15133-9</a>.</cite></li> <li><cite class="citation pubblicazione" style="font-style:normal"> Lucia Doretti, <a rel="nofollow" class="external text" href="https://www.dsv.unisi.it/sites/st15/files/allegatiparagrafo/01-12-2014/6._limiti_di_funzioni.pdf"><span style="font-style:italic;">Limiti di funzioni</span></a> (<span style="font-weight: bolder; font-size:80%"><abbr title="documento in formato PDF">PDF</abbr></span>), Dipartimento di Ingegneria dell'Informazione e Scienze Matematiche dell'<a href="/wiki/Universit%C3%A0_di_Siena" class="mw-redirect" title="Università di Siena">Università di Siena</a>.</cite></li> <li><a href="/wiki/Paolo_Marcellini" title="Paolo Marcellini">Paolo Marcellini</a>, <a href="/wiki/Carlo_Sbordone" title="Carlo Sbordone">Carlo Sbordone</a>, <i>Analisi Matematica Uno</i>, Liguori Editore, Napoli, <a href="/wiki/Speciale:RicercaISBN/8820728192" class="internal mw-magiclink-isbn">ISBN 88-207-2819-2</a>, 1998.</li> <li><a href="/wiki/Nicola_Fusco_(matematico)" title="Nicola Fusco (matematico)">Nicola Fusco</a>, <a href="/wiki/Paolo_Marcellini" title="Paolo Marcellini">Paolo Marcellini</a>, <a href="/wiki/Carlo_Sbordone" title="Carlo Sbordone">Carlo Sbordone</a>, <i>Lezioni di analisi matematica due</i>, Zanichelli Editore, Bologna, <a href="/wiki/Speciale:RicercaISBN/9788808520203" class="internal mw-magiclink-isbn">ISBN 978-88-08-52020-3</a>, 2020.</li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Moore E.H., Smith H.L., "A General Theory of Limits". <i>American Journal of Mathematics</i> <b>44</b> (2), 102–121 (1922).</li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Miller, N. <i>Limits: An Introductory Treatment</i>. Waltham, MA: Blaisdell, 1964.</li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Gruntz, D. <i>On Computing Limits in a Symbolic Manipulation System</i>. Doctoral thesis. Zürich: Swiss Federal Institute of Technology, 1996.</li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Hight, D. W. <i>A Concept of Limits</i>. New York: Prentice-Hall, 1966.</li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Kaplan, W. "Limits and Continuity." §2.4 in <i>Advanced Calculus, 4th ed</i>. Reading, MA: Addison-Wesley, pp. 82–86, 1992.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limite_(matematica)&veaction=edit&section=8" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Limite_(matematica)&action=edit&section=8" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Convergenza" title="Convergenza">Convergenza</a></li> <li><a href="/wiki/Forma_indeterminata" title="Forma indeterminata">Forma indeterminata</a></li> <li><a href="/wiki/Limite_di_una_funzione" title="Limite di una funzione">Limite di una funzione</a></li> <li><a href="/wiki/Limite_di_una_successione" title="Limite di una successione">Limite di una successione</a></li> <li><a href="/wiki/Limite_insiemistico" title="Limite insiemistico">Limite insiemistico</a></li> <li><a href="/wiki/Limite_notevole" title="Limite notevole">Limite notevole</a></li> <li><a href="/wiki/Limite_superiore_e_limite_inferiore" title="Limite superiore e limite inferiore">Limite superiore e limite inferiore</a></li> <li><a href="/wiki/Regola_di_De_L%27H%C3%B4pital" class="mw-redirect" title="Regola di De L'Hôpital">Regola di De L'Hôpital</a></li> <li><a href="/wiki/Stima_asintotica" title="Stima asintotica">Stima asintotica</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Altri_progetti">Altri progetti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limite_(matematica)&veaction=edit&section=9" title="Modifica la sezione Altri progetti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Limite_(matematica)&action=edit&section=9" title="Edit section's source code: Altri progetti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div id="interProject" class="toccolours" style="display: none; clear: both; margin-top: 2em"><p id="sisterProjects" style="background-color: #efefef; color: black; font-weight: bold; margin: 0"><span>Altri progetti</span></p><ul title="Collegamenti verso gli altri progetti Wikimedia"> <li class="" title=""><span class="plainlinks" title="commons:Category:Limit (mathematics)"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Limit_(mathematics)?uselang=it">Wikimedia Commons</a></span></li></ul></div> <ul><li><span typeof="mw:File"><a href="https://commons.wikimedia.org/wiki/?uselang=it" title="Collabora a Wikimedia Commons"><img alt="Collabora a Wikimedia Commons" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png" decoding="async" width="18" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/27px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/36px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/?uselang=it">Wikimedia Commons</a></span> contiene immagini o altri file sui <b><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Limit_(mathematics)?uselang=it">limiti</a></span></b></li></ul> <div class="mw-heading mw-heading2"><h2 id="Collegamenti_esterni">Collegamenti esterni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limite_(matematica)&veaction=edit&section=10" title="Modifica la sezione Collegamenti esterni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Limite_(matematica)&action=edit&section=10" title="Edit section's source code: Collegamenti esterni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li class="mw-empty-elt"></li> <li><cite id="CITEREFTreccani.it" class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/limite"><span style="font-style:italic;">limite</span></a>, su <span style="font-style:italic;">Treccani.it – Enciclopedie on line</span>, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q177239#P3365" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFBritannica.com" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://www.britannica.com/topic/limit-mathematics"><span style="font-style:italic;">limit</span></a>, su <span style="font-style:italic;"><a href="/wiki/Enciclopedia_Britannica" title="Enciclopedia Britannica">Enciclopedia Britannica</a></span>, Encyclopædia Britannica, Inc.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q177239#P1417" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFMathWorld" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Eric W. Weisstein, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Limit.html"><span style="font-style:italic;">Limite</span></a>, su <span style="font-style:italic;"><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></span>, Wolfram Research.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q177239#P2812" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite class="citation testo" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) L.D. Kudryavtsev, <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php/Limit"><span style="font-style:italic;">Limit</span></a>, in <span style="font-style:italic;"><a href="/wiki/Encyclopaedia_of_Mathematics" title="Encyclopaedia of Mathematics">Encyclopaedia of Mathematics</a></span>, Springer e European Mathematical Society, 2002.</cite></li></ul> <style data-mw-deduplicate="TemplateStyles:r141815314">.mw-parser-output .navbox{border:1px solid #aaa;clear:both;margin:auto;padding:2px;width:100%}.mw-parser-output .navbox th{padding-left:1em;padding-right:1em;text-align:center}.mw-parser-output .navbox>tbody>tr:first-child>th{background:#ccf;font-size:90%;width:100%;color:var(--color-base,black)}.mw-parser-output .navbox_navbar{float:left;margin:0;padding:0 10px 0 0;text-align:left;width:6em}.mw-parser-output .navbox_title{font-size:110%}.mw-parser-output .navbox_abovebelow{background:#ddf;font-size:90%;font-weight:normal}.mw-parser-output .navbox_group{background:#ddf;font-size:90%;padding:0 10px;white-space:nowrap}.mw-parser-output .navbox_list{font-size:90%;width:100%}.mw-parser-output .navbox_list a{white-space:nowrap}html:not(.vector-feature-night-mode-enabled) .mw-parser-output .navbox_odd{background:#fdfdfd;color:var(--color-base,black)}html:not(.vector-feature-night-mode-enabled) .mw-parser-output .navbox_even{background:#f7f7f7;color:var(--color-base,black)}.mw-parser-output .navbox a.mw-selflink{color:var(--color-base,black)}.mw-parser-output .navbox_center{text-align:center}.mw-parser-output .navbox .navbox_image{padding-left:7px;vertical-align:middle;width:0}.mw-parser-output .navbox+.navbox{margin-top:-1px}.mw-parser-output .navbox .mw-collapsible-toggle{font-weight:normal;text-align:right;width:7em}body.skin--responsive .mw-parser-output .navbox_image img{max-width:none!important}.mw-parser-output .subnavbox{margin:-3px;width:100%}.mw-parser-output .subnavbox_group{background:#e6e6ff;padding:0 10px}@media screen{html.skin-theme-clientpref-night .mw-parser-output .navbox>tbody>tr:first-child>th{background:var(--background-color-interactive)!important}html.skin-theme-clientpref-night .mw-parser-output .navbox th{color:var(--color-base)!important}html.skin-theme-clientpref-night .mw-parser-output .navbox_abovebelow,html.skin-theme-clientpref-night .mw-parser-output .navbox_group{background:var(--background-color-interactive-subtle)!important}html.skin-theme-clientpref-night .mw-parser-output .subnavbox_group{background:var(--background-color-neutral-subtle)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbox>tbody>tr:first-child>th{background:var(--background-color-interactive)!important}html.skin-theme-clientpref-os .mw-parser-output .navbox th{color:var(--color-base)!important}html.skin-theme-clientpref-os .mw-parser-output .navbox_abovebelow,html.skin-theme-clientpref-os .mw-parser-output .navbox_group{background:var(--background-color-interactive-subtle)!important}html.skin-theme-clientpref-os .mw-parser-output .subnavbox_group{background:var(--background-color-neutral-subtle)!important}}</style><table class="navbox mw-collapsible mw-collapsed noprint metadata" id="navbox-Analisi_matematica"><tbody><tr><th colspan="3"><div class="navbox_navbar"><div class="noprint plainlinks" style="background-color:transparent; padding:0; font-size:xx-small; color:var(--color-base, #000000); white-space:nowrap;"><a href="/wiki/Template:Analisi_matematica" title="Template:Analisi matematica"><span title="Vai alla pagina del template">V</span></a> · <a href="/w/index.php?title=Discussioni_template:Analisi_matematica&action=edit&redlink=1" class="new" title="Discussioni template:Analisi matematica (la pagina non esiste)"><span title="Discuti del template">D</span></a> · <a class="external text" href="https://it.wikipedia.org/w/index.php?title=Template:Analisi_matematica&action=edit"><span title="Modifica il template. Usa l'anteprima prima di salvare">M</span></a></div></div><span class="navbox_title"><a href="/wiki/Analisi_matematica" title="Analisi matematica">Analisi matematica</a></span></th></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Calcolo_infinitesimale" title="Calcolo infinitesimale">Calcolo infinitesimale</a></th><td colspan="1" class="navbox_list navbox_odd"><a href="/wiki/Numero_reale" title="Numero reale">Numero reale</a><b> ·</b> <a href="/wiki/Infinitesimo" title="Infinitesimo">Infinitesimo</a><b> ·</b> <a href="/wiki/O-grande" title="O-grande">O-grande</a><b> ·</b> <a href="/wiki/Successione_(matematica)" title="Successione (matematica)">Successione</a> (<a href="/wiki/Successione_di_funzioni" title="Successione di funzioni">di funzioni</a>)<b> ·</b> <a href="/wiki/Successione_di_Cauchy" title="Successione di Cauchy">Successione di Cauchy</a><b> ·</b> <a href="/wiki/Teorema_di_Bolzano-Weierstrass" title="Teorema di Bolzano-Weierstrass">Teorema di Bolzano-Weierstrass</a><b> ·</b> <a href="/wiki/Stima_asintotica" title="Stima asintotica">Stima asintotica</a><b> ·</b> <a class="mw-selflink selflink">Limite</a> (<a href="/wiki/Limite_di_una_funzione" title="Limite di una funzione">di una funzione</a><b> ·</b> <a href="/wiki/Limite_di_una_successione" title="Limite di una successione">di una successione</a><b> ·</b> <a href="/wiki/Forma_indeterminata" title="Forma indeterminata">Forma indeterminata</a>)<b> ·</b> <a href="/wiki/Teorema_del_confronto" title="Teorema del confronto">Teorema dei due carabinieri</a><b> ·</b> <a href="/wiki/Limite_notevole" title="Limite notevole">Limite notevole</a><b> ·</b> <a href="/wiki/Punto_di_accumulazione" title="Punto di accumulazione">Punto di accumulazione</a><b> ·</b> <a href="/wiki/Punto_isolato" title="Punto isolato">Punto isolato</a><b> ·</b> <a href="/wiki/Intorno" title="Intorno">Intorno</a><b> ·</b> <a href="/wiki/Serie_(matematica)" title="Serie (matematica)">Serie</a> (<a href="/wiki/Serie_di_funzioni" title="Serie di funzioni">di funzioni</a>)<b> ·</b> <a href="/wiki/Criteri_di_convergenza" title="Criteri di convergenza">Criteri di convergenza</a><b> ·</b> <a href="/wiki/Limite_di_funzioni_a_pi%C3%B9_variabili" title="Limite di funzioni a più variabili">Limite di funzioni a più variabili</a></td><td rowspan="7" class="navbox_image"><figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_kmplot.svg" class="mw-file-description" title="Analisi matematica"><img alt="Analisi matematica" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Nuvola_apps_kmplot.svg/100px-Nuvola_apps_kmplot.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Nuvola_apps_kmplot.svg/150px-Nuvola_apps_kmplot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Nuvola_apps_kmplot.svg/200px-Nuvola_apps_kmplot.svg.png 2x" data-file-width="400" data-file-height="400" /></a><figcaption>Analisi matematica</figcaption></figure></td></tr><tr><th colspan="1" class="navbox_group">Studio della continuità</th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Funzione_continua" title="Funzione continua">Funzione continua</a><b> ·</b> <a href="/wiki/Punto_di_discontinuit%C3%A0" title="Punto di discontinuità">Punto di discontinuità</a><b> ·</b> <a href="/wiki/Continuit%C3%A0_uniforme" title="Continuità uniforme">Continuità uniforme</a><b> ·</b> <a href="/wiki/Funzione_lipschitziana" title="Funzione lipschitziana">Funzione lipschitziana</a><b> ·</b> <a href="/wiki/Teorema_di_Bolzano" title="Teorema di Bolzano">Teorema di Bolzano</a><b> ·</b> <a href="/wiki/Teorema_di_Weierstrass" title="Teorema di Weierstrass">Teorema di Weierstrass</a><b> ·</b> <a href="/wiki/Teorema_dei_valori_intermedi" title="Teorema dei valori intermedi">Teorema dei valori intermedi</a><b> ·</b> <a href="/wiki/Teorema_di_Heine-Cantor" title="Teorema di Heine-Cantor">Teorema di Heine-Cantor</a><b> ·</b> <a href="/wiki/Modulo_di_continuit%C3%A0" title="Modulo di continuità">Modulo di continuità</a><b> ·</b> <a href="/wiki/Funzione_semicontinua" class="mw-redirect" title="Funzione semicontinua">Funzione semicontinua</a><b> ·</b> <a href="/wiki/Continuit%C3%A0_separata" title="Continuità separata">Continuità separata</a><b> ·</b> <a href="/wiki/Teorema_di_approssimazione_di_Weierstrass" title="Teorema di approssimazione di Weierstrass">Teorema di approssimazione di Weierstrass</a></td></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Calcolo_differenziale" class="mw-redirect" title="Calcolo differenziale">Calcolo differenziale</a></th><td colspan="1" class="navbox_list navbox_odd"><a href="/wiki/Derivata" title="Derivata">Derivata</a><b> ·</b> <a href="/wiki/Differenziale_(matematica)" title="Differenziale (matematica)">Differenziale</a><b> ·</b> <a href="/wiki/Regole_di_derivazione" title="Regole di derivazione">Regole di derivazione</a><b> ·</b> <a href="/wiki/Teorema_di_Fermat_sui_punti_stazionari" title="Teorema di Fermat sui punti stazionari">Teorema di Fermat</a><b> ·</b> <a href="/wiki/Teorema_di_Rolle" title="Teorema di Rolle">Teorema di Rolle</a><b> ·</b> <a href="/wiki/Teorema_di_Lagrange" title="Teorema di Lagrange">Teorema di Lagrange</a><b> ·</b> <a href="/wiki/Teorema_di_Cauchy_(analisi_matematica)" title="Teorema di Cauchy (analisi matematica)">Teorema di Cauchy</a><b> ·</b> <a href="/wiki/Teorema_di_Darboux" title="Teorema di Darboux">Teorema di Darboux</a><b> ·</b> <a href="/wiki/Teorema_di_Taylor" title="Teorema di Taylor">Teorema di Taylor</a><b> ·</b> <a href="/wiki/Serie_di_Taylor" title="Serie di Taylor">Serie di Taylor</a><b> ·</b> <a href="/wiki/Funzione_differenziabile" title="Funzione differenziabile">Funzione differenziabile</a><b> ·</b> <a href="/wiki/Gradiente_(funzione)" class="mw-redirect" title="Gradiente (funzione)">Gradiente</a><b> ·</b> <a href="/wiki/Matrice_jacobiana" title="Matrice jacobiana">Jacobiana</a><b> ·</b> <a href="/wiki/Matrice_hessiana" title="Matrice hessiana">Hessiana</a><b> ·</b> <a href="/wiki/Forma_differenziale" title="Forma differenziale">Forma differenziale</a><b> ·</b> <a href="/wiki/Generalizzazioni_della_derivata" title="Generalizzazioni della derivata">Generalizzazioni della derivata</a><b> ·</b> <a href="/wiki/Derivata_parziale" title="Derivata parziale">Derivata parziale</a><b> ·</b> <a href="/wiki/Derivata_mista" title="Derivata mista">Derivata mista</a></td></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Integrale" title="Integrale">Integrale</a></th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Primitiva_(matematica)" title="Primitiva (matematica)">Primitiva</a><b> ·</b> <a href="/wiki/Integrale_di_Riemann" title="Integrale di Riemann">Integrale di Riemann</a><b> ·</b> <a href="/wiki/Integrale_improprio" title="Integrale improprio">Integrale improprio</a><b> ·</b> <a href="/wiki/Integrale_di_Lebesgue" title="Integrale di Lebesgue">Integrale di Lebesgue</a><b> ·</b> <a href="/wiki/Teorema_fondamentale_del_calcolo_integrale" title="Teorema fondamentale del calcolo integrale">Teorema fondamentale</a><b> ·</b> <a href="/wiki/Metodi_di_integrazione" title="Metodi di integrazione">Metodi di integrazione</a><b> ·</b> <a href="/wiki/Categoria:Tavole_di_integrali" title="Categoria:Tavole di integrali">Tavole</a><b> ·</b> <a href="/wiki/Integrale_multiplo" title="Integrale multiplo">Integrale multiplo</a>, <a href="/wiki/Integrale_di_linea" title="Integrale di linea">di linea</a> (<a href="/wiki/Integrale_di_linea_di_prima_specie" title="Integrale di linea di prima specie">1ª specie</a><b> ·</b> <a href="/wiki/Integrale_di_linea_di_seconda_specie" title="Integrale di linea di seconda specie">2ª specie</a>) e <a href="/wiki/Integrale_di_superficie" title="Integrale di superficie">di superficie</a> (<a href="/wiki/Integrale_di_volume" title="Integrale di volume">di volume</a>)</td></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Studio_di_funzione" title="Studio di funzione">Studio di funzione</a></th><td colspan="1" class="navbox_list navbox_odd"><a href="/wiki/Funzione_(matematica)" title="Funzione (matematica)">Funzione</a><b> ·</b> <a href="/wiki/Variabile_(matematica)" title="Variabile (matematica)">Variabile</a><b> ·</b> <a href="/wiki/Dominio_e_codominio" title="Dominio e codominio">Dominio e codominio</a><b> ·</b> <a href="/wiki/Funzioni_pari_e_dispari" title="Funzioni pari e dispari">Funzioni pari e dispari</a><b> ·</b> <a href="/wiki/Funzione_periodica" title="Funzione periodica">Funzione periodica</a><b> ·</b> <a href="/wiki/Funzione_monotona" title="Funzione monotona">Funzione monotona</a><b> ·</b> <a href="/wiki/Funzione_convessa" title="Funzione convessa">Funzione convessa</a><b> ·</b> <a href="/wiki/Massimo_e_minimo_di_una_funzione" title="Massimo e minimo di una funzione">Massimo e minimo di una funzione</a><b> ·</b> <a href="/wiki/Punto_angoloso" title="Punto angoloso">Punto angoloso</a><b> ·</b> <a href="/wiki/Cuspide_(matematica)" title="Cuspide (matematica)">Cuspide</a><b> ·</b> <a href="/wiki/Punto_di_flesso" title="Punto di flesso">Punto di flesso</a><b> ·</b> <a href="/wiki/Asintoto" title="Asintoto">Asintoto</a><b> ·</b> <a href="/wiki/Grafico_di_una_funzione" title="Grafico di una funzione">Grafico di una funzione</a><b> ·</b> <a href="/wiki/Funzione_iniettiva" title="Funzione iniettiva">Funzione iniettiva</a></td></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Disuguaglianza" title="Disuguaglianza">Disuguaglianze</a></th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Disuguaglianza_triangolare" title="Disuguaglianza triangolare">Disuguaglianza triangolare</a><b> ·</b> <a href="/wiki/Disuguaglianza_di_Cauchy-Schwarz" title="Disuguaglianza di Cauchy-Schwarz">Disuguaglianza di Cauchy-Schwarz</a><b> ·</b> <a href="/wiki/Disuguaglianza_di_Bernoulli" title="Disuguaglianza di Bernoulli">Bernoulli</a><b> ·</b> <a href="/wiki/Disuguaglianza_di_Jensen" title="Disuguaglianza di Jensen">Jensen</a><b> ·</b> <a href="/wiki/Disuguaglianza_di_H%C3%B6lder" title="Disuguaglianza di Hölder">Hölder</a><b> ·</b> <a href="/wiki/Disuguaglianza_di_Young" title="Disuguaglianza di Young">Young</a><b> ·</b> <a href="/wiki/Disuguaglianza_di_Minkowski" title="Disuguaglianza di Minkowski">Minkowski</a></td></tr><tr><th colspan="1" class="navbox_group">Altro</th><td colspan="1" class="navbox_list navbox_odd"><a href="/wiki/Approssimazione_di_Stirling" title="Approssimazione di Stirling">Approssimazione di Stirling</a><b> ·</b> <a href="/wiki/Prodotto_di_Wallis" title="Prodotto di Wallis">Prodotto di Wallis</a><b> ·</b> <a href="/wiki/Funzione_Gamma" title="Funzione Gamma">Funzione Gamma</a><b> ·</b> <a href="/wiki/Teorema_delle_funzioni_implicite" title="Teorema delle funzioni implicite">Teorema delle funzioni implicite</a><b> ·</b> <a href="/wiki/Teorema_della_funzione_inversa" title="Teorema della funzione inversa">Teorema della funzione inversa</a><b> ·</b> <a href="/wiki/Condizione_di_H%C3%B6lder" title="Condizione di Hölder">Funzione hölderiana</a><b> ·</b> <a href="/wiki/Spazio_metrico" title="Spazio metrico">Spazio metrico</a><b> ·</b> <a href="/wiki/Spazio_normato" title="Spazio normato">Spazio normato</a><b> ·</b> <a href="/wiki/Intervallo_(matematica)" title="Intervallo (matematica)">Intervallo</a><b> ·</b> <a href="/wiki/Insieme_trascurabile" title="Insieme trascurabile">Insieme trascurabile</a><b> ·</b> <a href="/wiki/Insieme_chiuso" title="Insieme chiuso">Insieme chiuso</a><b> ·</b> <a href="/wiki/Insieme_aperto" title="Insieme aperto">Insieme aperto</a><b> ·</b> <a href="/wiki/Palla_(matematica)" title="Palla (matematica)">Palla</a><b> ·</b> <a href="/wiki/Omeomorfismo" title="Omeomorfismo">Omeomorfismo</a><b> ·</b> <a href="/wiki/Omeomorfismo_locale" title="Omeomorfismo locale">Omeomorfismo locale</a><b> ·</b> <a href="/wiki/Diffeomorfismo" title="Diffeomorfismo">Diffeomorfismo</a><b> ·</b> <a href="/wiki/Diffeomorfismo_locale" title="Diffeomorfismo locale">Diffeomorfismo locale</a><b> ·</b> <a href="/wiki/Classe_C_di_una_funzione" title="Classe C di una funzione">Classe C di una funzione</a><b> ·</b> <a href="/wiki/Equazione_differenziale" title="Equazione differenziale">Equazione differenziale</a><b> ·</b> <a href="/wiki/Problema_di_Cauchy" title="Problema di Cauchy">Problema di Cauchy</a></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r141815314"><table class="navbox mw-collapsible mw-collapsed noprint metadata" id="navbox-Topologia"><tbody><tr><th colspan="3"><div class="navbox_navbar"><div class="noprint plainlinks" style="background-color:transparent; padding:0; font-size:xx-small; color:var(--color-base, #000000); white-space:nowrap;"><a href="/wiki/Template:Topologia" title="Template:Topologia"><span title="Vai alla pagina del template">V</span></a> · <a href="/w/index.php?title=Discussioni_template:Topologia&action=edit&redlink=1" class="new" title="Discussioni template:Topologia (la pagina non esiste)"><span title="Discuti del template">D</span></a> · <a class="external text" href="https://it.wikipedia.org/w/index.php?title=Template:Topologia&action=edit"><span title="Modifica il template. Usa l'anteprima prima di salvare">M</span></a></div></div><span class="navbox_title"><a href="/wiki/Topologia" title="Topologia">Topologia</a></span></th></tr><tr><th colspan="1" class="navbox_group" style="text-align:right;">Concetti di <a href="/wiki/Topologia_generale" title="Topologia generale">Topologia generale</a></th><td colspan="1" class="navbox_list navbox_odd"><table class="subnavbox"><tbody><tr><td colspan="2" class="navbox_center"><a href="/wiki/Spazio_topologico" title="Spazio topologico">Spazio topologico</a><b> ·</b> <a href="/wiki/Base_(topologia)" title="Base (topologia)">Base</a><b> ·</b> <a href="/wiki/Prebase" title="Prebase">Prebase</a><b> ·</b> <a href="/wiki/Ricoprimento" title="Ricoprimento">Ricoprimento</a><b> ·</b> <a href="/wiki/Assiomi_di_chiusura_di_Kuratowski" title="Assiomi di chiusura di Kuratowski">Assiomi di chiusura di Kuratowski</a><b> ·</b> <a href="/wiki/Invariante_topologico" title="Invariante topologico">Invariante topologico</a><b> ·</b> <a href="/wiki/Relazione_di_finezza" title="Relazione di finezza">Relazione di finezza</a><b> ·</b> <a href="/wiki/Partizione_dell%27unit%C3%A0" title="Partizione dell'unità">Partizione dell'unità</a><b> ·</b> <a href="/wiki/Propriet%C3%A0_dell%27intersezione_finita" title="Proprietà dell'intersezione finita">Proprietà dell'intersezione finita</a></td></tr><tr><th class="subnavbox_group">Sottoinsiemi</th><td colspan="1"><a href="/wiki/Intervallo_(matematica)" title="Intervallo (matematica)">Intervallo</a><b> ·</b> <a href="/wiki/Insieme_aperto" title="Insieme aperto">Aperto</a><b> ·</b> <a href="/wiki/Intorno" title="Intorno">Intorno</a><b> ·</b> <a href="/wiki/Insieme_chiuso" title="Insieme chiuso">Chiuso</a><b> ·</b> <a href="/wiki/Insieme_localmente_chiuso" title="Insieme localmente chiuso">Insieme localmente chiuso</a><b> ·</b> <a href="/wiki/Insieme_chiuso-aperto" title="Insieme chiuso-aperto">Insieme chiuso-aperto</a><b> ·</b> <a href="/wiki/Parte_interna" title="Parte interna">Parte interna</a><b> ·</b> <a href="/wiki/Chiusura_(topologia)" title="Chiusura (topologia)">Chiusura</a><b> ·</b> <a href="/wiki/Frontiera_(topologia)" title="Frontiera (topologia)">Frontiera</a><b> ·</b> <a href="/wiki/Insieme_derivato" title="Insieme derivato">Insieme derivato</a><b> ·</b> <a href="/wiki/Insieme_limite" title="Insieme limite">Insieme limite</a><b> ·</b> <a href="/wiki/Insieme_perfetto" title="Insieme perfetto">Insieme perfetto</a><b> ·</b> <a href="/wiki/Insieme_denso" title="Insieme denso">Insieme denso</a><b> ·</b> <a href="/wiki/Insieme_mai_denso" title="Insieme mai denso">Insieme mai denso</a></td></tr><tr><th class="subnavbox_group">Punti</th><td colspan="1"><a href="/wiki/Punto_isolato" title="Punto isolato">Punto isolato</a><b> ·</b> <a href="/wiki/Punto_di_accumulazione" title="Punto di accumulazione">Punto di accumulazione</a><b> ·</b> <a href="/wiki/Punto_di_aderenza" title="Punto di aderenza">Punto di aderenza</a></td></tr><tr><th class="subnavbox_group">Funzioni</th><td colspan="1"><a href="/wiki/Funzione_continua" title="Funzione continua">Funzione continua</a><b> ·</b> <a href="/wiki/Omeomorfismo" title="Omeomorfismo">Omeomorfismo</a><b> ·</b> <a href="/wiki/Funzione_aperta" title="Funzione aperta">Funzione aperta</a><b> ·</b> <a href="/wiki/Funzione_chiusa" title="Funzione chiusa">Funzione chiusa</a><b> ·</b> <a href="/wiki/Funzione_propria" title="Funzione propria">Funzione propria</a><b> ·</b> <a href="/wiki/Contrazione_(matematica)" title="Contrazione (matematica)">Contrazione</a><b> ·</b> <a href="/wiki/Retrazione" title="Retrazione">Retrazione</a><b> ·</b> <a href="/wiki/Germe_di_funzione" title="Germe di funzione">Germe di funzione</a><b> ·</b> <a href="/wiki/Funzione_a_supporto_compatto" title="Funzione a supporto compatto">Funzione a supporto compatto</a></td></tr><tr><th class="subnavbox_group">Successioni</th><td colspan="1"><a class="mw-selflink selflink">Limite</a><b> ·</b> <a href="/wiki/Limite_di_una_successione" title="Limite di una successione">Limite di una successione</a><b> ·</b> <a href="/wiki/Successione_(matematica)" title="Successione (matematica)">Successione</a><b> ·</b> <a href="/wiki/Rete_(matematica)" title="Rete (matematica)">Rete</a><b> ·</b> <a href="/wiki/Convergenza" title="Convergenza">Convergenza</a><b> ·</b> <a href="/wiki/Successione_di_Cauchy" title="Successione di Cauchy">Successione di Cauchy</a></td></tr><tr><th class="subnavbox_group">Teoremi</th><td colspan="1"><a href="/wiki/Teorema_di_Weierstrass" title="Teorema di Weierstrass">Teorema di Weierstrass</a><b> ·</b> <a href="/wiki/Teorema_di_Heine-Borel" title="Teorema di Heine-Borel">Heine-Borel</a><b> ·</b> <a href="/wiki/Teorema_di_Tichonov" title="Teorema di Tichonov">Tichonov</a><b> ·</b> <a href="/wiki/Lemma_del_tubo" title="Lemma del tubo">Lemma del tubo</a><b> ·</b> <a href="/wiki/Lemma_di_Urysohn" title="Lemma di Urysohn">Urysohn</a><b> ·</b> <a href="/wiki/Teorema_di_estensione_di_Tietze" title="Teorema di estensione di Tietze">Tietze</a><b> ·</b> <a href="/wiki/Teorema_della_categoria_di_Baire" title="Teorema della categoria di Baire">Baire</a><b> ·</b> <a href="/wiki/Teorema_del_punto_fisso_di_Brouwer" title="Teorema del punto fisso di Brouwer">Brouwer</a><b> ·</b> <a href="/wiki/Teoremi_di_punto_fisso" title="Teoremi di punto fisso">punto fisso</a><b> ·</b> <a href="/wiki/Teorema_di_Borsuk" title="Teorema di Borsuk">Teorema di Borsuk</a><b> ·</b> <a href="/wiki/Teorema_di_Borsuk-Ulam" title="Teorema di Borsuk-Ulam">Teorema di Borsuk-Ulam</a><b> ·</b> <a href="/wiki/Teorema_della_curva_di_Jordan" title="Teorema della curva di Jordan">Teorema della curva di Jordan</a><b> ·</b> <a href="/wiki/Teorema_della_mappa_di_Riemann" title="Teorema della mappa di Riemann">Teorema della mappa di Riemann</a></td></tr><tr><th class="subnavbox_group">Applicazioni pratiche</th><td colspan="1"><a href="/wiki/Topologia_dello_spazio-tempo" title="Topologia dello spazio-tempo">Topologia dello spazio-tempo</a><b> ·</b> <a href="/wiki/Teoria_quantistica_dei_campi_topologica" title="Teoria quantistica dei campi topologica">Teoria quantistica dei campi topologica</a><b> ·</b> <a href="/wiki/K-teoria_ritorta" title="K-teoria ritorta">K-teoria ritorta</a><b> ·</b> <a href="/wiki/Topologia_di_rete" title="Topologia di rete">Topologia di rete</a><b> ·</b> <a href="/wiki/Controllo_della_topologia" title="Controllo della topologia">Controllo della topologia</a><b> ·</b> <a href="/wiki/Topologia_molecolare" title="Topologia molecolare">Topologia molecolare</a></td></tr></tbody></table></td><td rowspan="6" class="navbox_image"><span typeof="mw:File"><a href="/wiki/File:Torus.jpg" class="mw-file-description" title="Toro"><img alt="Toro" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Torus.jpg/100px-Torus.jpg" decoding="async" width="100" height="56" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Torus.jpg/150px-Torus.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/Torus.jpg/200px-Torus.jpg 2x" data-file-width="300" data-file-height="168" /></a></span></td></tr><tr><th colspan="1" class="navbox_group" style="text-align:right;">Spazi topologici</th><td colspan="1" class="navbox_list navbox_even"><table class="subnavbox"><tbody><tr><th class="subnavbox_group">Topologie classiche</th><td colspan="1"><a href="/wiki/Topologia_banale" title="Topologia banale">Topologia banale</a><b> ·</b> <a href="/wiki/Spazio_di_Sierpi%C5%84ski" title="Spazio di Sierpiński">Spazio di Sierpiński</a><b> ·</b> <a href="/wiki/Topologia_cofinita" title="Topologia cofinita">Cofinita</a><b> ·</b> <a href="/wiki/Topologia_della_semicontinuit%C3%A0_inferiore" title="Topologia della semicontinuità inferiore">Topologia della semicontinuità inferiore</a><b> ·</b> <a href="/wiki/Topologia_di_Zariski" title="Topologia di Zariski">di Zariski</a><b> ·</b> <a href="/wiki/Spazio_euclideo#Topologia_euclidea" title="Spazio euclideo">Euclidea</a><b> ·</b> <a href="/wiki/Topologia_del_limite_inferiore" title="Topologia del limite inferiore">del limite inferiore</a> o di Sorgenfrey<b> ·</b> <a href="/wiki/Topologia_discreta" title="Topologia discreta">Discreta</a><b> ·</b> <a href="/wiki/Topologia_degli_interi_equispaziati" title="Topologia degli interi equispaziati">Topologia degli interi equispaziati</a><b> ·</b> <a href="/wiki/Numero_reale#Insieme_reale_esteso" title="Numero reale">Insieme reale esteso</a><b> ·</b> <a href="/wiki/Ordine_totale#Topologia_di_ordine" title="Ordine totale">Topologia di ordine</a><b> ·</b> <a href="/wiki/Piano_di_Moore" title="Piano di Moore">Piano di Moore</a><b> ·</b> <a href="/wiki/Numero_p-adico" title="Numero p-adico">Topologia p-adica</a></td></tr><tr><th class="subnavbox_group">Costruzioni topologiche</th><td colspan="1"><a href="/wiki/Topologia_prodotto" title="Topologia prodotto">Topologia prodotto</a><b> ·</b> <a href="/wiki/Topologia_di_sottospazio" title="Topologia di sottospazio">Topologia di sottospazio</a><b> ·</b> <a href="/wiki/Topologia_quoziente" title="Topologia quoziente">Topologia quoziente</a><b> ·</b> <a href="/wiki/Compattificazione" title="Compattificazione">Compattificazione</a> (<a href="/wiki/Compattificazione_di_Alexandrov" title="Compattificazione di Alexandrov">di Alexandrov</a><b> ·</b> <a href="/wiki/Compattificazione_di_Stone-%C4%8Cech" title="Compattificazione di Stone-Čech">di Stone-Čech</a>)<b> ·</b> <a href="/wiki/Cono_(topologia)" title="Cono (topologia)">Cono</a><b> ·</b> <a href="/wiki/Bouquet_(topologia)" title="Bouquet (topologia)">Bouquet</a><b> ·</b> <a href="/wiki/Rosa_(topologia)" title="Rosa (topologia)">Rosa</a><b> ·</b> <a href="/wiki/Sospensione_(matematica)" title="Sospensione (matematica)">Sospensione</a></td></tr><tr><th class="subnavbox_group">Topologie in <a href="/wiki/Analisi_funzionale" title="Analisi funzionale">Analisi funzionale</a></th><td colspan="1"><a href="/wiki/Spazio_funzionale" title="Spazio funzionale">Spazio funzionale</a><b> ·</b> <a href="/wiki/Topologia_iniziale" title="Topologia iniziale">Topologia iniziale</a> o debole<b> ·</b> <a href="/wiki/Topologia_operatoriale" title="Topologia operatoriale">Topologia operatoriale</a><b> ·</b> <a href="/wiki/Topologia_finale" title="Topologia finale">Topologia finale</a> o forte<b> ·</b> <a href="/wiki/Topologia_di_Mackey" title="Topologia di Mackey">Topologia di Mackey</a><b> ·</b> <a href="/wiki/Topologia_polare" title="Topologia polare">Topologia polare</a><b> ·</b> <a href="/wiki/Topologie_operatoriali_debole_e_forte" title="Topologie operatoriali debole e forte">Topologie operatoriali debole e forte</a></td></tr><tr><th class="subnavbox_group">Altri oggetti topologici</th><td colspan="1"><a href="/wiki/Sfera" title="Sfera">Sfera</a><b> ·</b> <a href="/wiki/Palla_(matematica)" title="Palla (matematica)">Palla</a><b> ·</b> <a href="/wiki/Toro_(geometria)" title="Toro (geometria)">Toro</a><b> ·</b> <a href="/wiki/Corpo_con_manici" title="Corpo con manici">Corpo con manici</a><b> ·</b> <a href="/wiki/Bottiglia_di_Klein" title="Bottiglia di Klein">Bottiglia di Klein</a><b> ·</b> <a href="/wiki/Bottiglia_di_Klein_solida" title="Bottiglia di Klein solida">Bottiglia di Klein solida</a><b> ·</b> <a href="/wiki/Anello_(topologia)" title="Anello (topologia)">Anello</a><b> ·</b> <a href="/wiki/Nastro_di_M%C3%B6bius" title="Nastro di Möbius">Nastro di Möbius</a><b> ·</b> <a href="/wiki/Retta_proiettiva" title="Retta proiettiva">Retta proiettiva</a><b> ·</b> <a href="/wiki/Piano_proiettivo" title="Piano proiettivo">Piano proiettivo</a><b> ·</b> <a href="/wiki/Superficie_di_Riemann" title="Superficie di Riemann">Superficie di Riemann</a><b> ·</b> <a href="/wiki/Teoria_dei_nodi" title="Teoria dei nodi">Nodo</a><b> ·</b> <a href="/wiki/Nodo_torico" title="Nodo torico">Nodo torico</a><b> ·</b> <a href="/wiki/Link_(teoria_dei_nodi)" title="Link (teoria dei nodi)">Link</a> <table class="subnavbox"><tbody><tr><th class="subnavbox_group"><a href="/wiki/Frattale" title="Frattale">Frattali</a></th><td colspan="1"><a href="/wiki/Insieme_di_Cantor" title="Insieme di Cantor">Insieme di Cantor</a><b> ·</b> <a href="/wiki/Spazio_di_Cantor" title="Spazio di Cantor">Spazio di Cantor</a><b> ·</b> <a href="/wiki/Polvere_di_Cantor" title="Polvere di Cantor">Polvere di Cantor</a><b> ·</b> <a href="/wiki/Spugna_di_Menger" title="Spugna di Menger">Spugna di Menger</a><b> ·</b> <a href="/wiki/Sfera_di_Alexander" title="Sfera di Alexander">Sfera di Alexander</a><b> ·</b> <a href="/wiki/Curva_di_Peano" title="Curva di Peano">Curva di Peano</a><b> ·</b> <a href="/wiki/Laghi_di_Wada" title="Laghi di Wada">Laghi di Wada</a></td></tr></tbody></table></td></tr><tr><th class="subnavbox_group">Strutture miste</th><td colspan="1"><a href="/wiki/Spazio_vettoriale_topologico" title="Spazio vettoriale topologico">Spazio vettoriale topologico</a><b> ·</b> <a href="/wiki/Gruppo_topologico" title="Gruppo topologico">Gruppo topologico</a><b> ·</b> <a href="/wiki/Gruppo_di_Lie" title="Gruppo di Lie">Gruppo di Lie</a><b> ·</b> <a href="/wiki/Spazio_uniforme" title="Spazio uniforme">Spazio uniforme</a><b> ·</b> <a href="/wiki/Algebra_di_Borel" title="Algebra di Borel">Algebra di Borel</a></td></tr></tbody></table></td></tr><tr><th colspan="1" class="navbox_group" style="text-align:right;"><a href="/wiki/Propriet%C3%A0_(matematica)" title="Proprietà (matematica)">Proprietà</a> degli spazi topologici</th><td colspan="1" class="navbox_list navbox_odd"><table class="subnavbox"><tbody><tr><th class="subnavbox_group">Numerabilità</th><td colspan="1"><a href="/wiki/Assioma_di_numerabilit%C3%A0" title="Assioma di numerabilità">Assioma di numerabilità</a><b> ·</b> <a href="/wiki/Spazio_primo-numerabile" title="Spazio primo-numerabile">Spazio primo-numerabile</a><b> ·</b> <a href="/wiki/Spazio_separabile" title="Spazio separabile">Spazio separabile</a><b> ·</b> <a href="/wiki/Spazio_sequenziale" title="Spazio sequenziale">Spazio sequenziale</a></td></tr><tr><th class="subnavbox_group">Separazione</th><td colspan="1"><a href="/wiki/Assioma_di_separazione" title="Assioma di separazione">Assioma di separazione</a><b> ·</b> <a href="/wiki/Spazio_T0" title="Spazio T0">Spazio T0</a><b> ·</b> <a href="/wiki/Spazio_T1" title="Spazio T1">Spazio T1</a><b> ·</b> <a href="/wiki/Spazio_di_Hausdorff" title="Spazio di Hausdorff">Spazio di Hausdorff</a><b> ·</b> <a href="/wiki/Spazio_regolare" title="Spazio regolare">Spazio regolare</a><b> ·</b> <a href="/wiki/Spazio_di_Tichonov" title="Spazio di Tichonov">Spazio di Tichonov</a><b> ·</b> <a href="/wiki/Spazio_normale" title="Spazio normale">Spazio normale</a></td></tr><tr><th class="subnavbox_group">Compattezza</th><td colspan="1"><a href="/wiki/Spazio_compatto" title="Spazio compatto">Spazio compatto</a><b> ·</b> <a href="/wiki/Spazio_paracompatto" title="Spazio paracompatto">Spazio paracompatto</a><b> ·</b> <a href="/wiki/Spazio_localmente_compatto" title="Spazio localmente compatto">Spazio localmente compatto</a><b> ·</b> <a href="/wiki/Spazio_di_Lindel%C3%B6f" title="Spazio di Lindelöf">Spazio di Lindelöf</a><b> ·</b> <a href="/wiki/Sottospazio_relativamente_compatto" title="Sottospazio relativamente compatto">Sottospazio relativamente compatto</a><b> ·</b> <a href="/wiki/Immersione_compatta" title="Immersione compatta">Immersione compatta</a></td></tr><tr><th class="subnavbox_group">Connessione</th><td colspan="1"><a href="/wiki/Spazio_connesso" title="Spazio connesso">Spazio connesso</a><b> ·</b> <a href="/wiki/Spazio_semplicemente_connesso" title="Spazio semplicemente connesso">Spazio semplicemente connesso</a></td></tr><tr><th class="subnavbox_group">Metrizzabilità</th><td colspan="1"><a href="/wiki/Spazio_metrico" title="Spazio metrico">Spazio metrico</a><b> ·</b> <a href="/wiki/Spazio_metrico_completo" title="Spazio metrico completo">Spazio metrico completo</a><b> ·</b> <a href="/wiki/Spazio_metrizzabile" title="Spazio metrizzabile">Spazio metrizzabile</a><b> ·</b> <a href="/wiki/Spazio_ultrametrico" title="Spazio ultrametrico">Spazio ultrametrico</a><b> ·</b> <a href="/wiki/Spazio_pseudometrico" title="Spazio pseudometrico">Spazio pseudometrico</a><b> ·</b> <a href="/wiki/Spazio_polacco" title="Spazio polacco">Spazio polacco</a><b> ·</b> <a href="/wiki/Spazio_normato" title="Spazio normato">Spazio normato</a><b> ·</b> <a href="/wiki/Spazio_totalmente_limitato" title="Spazio totalmente limitato">Spazio totalmente limitato</a></td></tr><tr><th class="subnavbox_group">Altre proprietà</th><td colspan="1"><a href="/wiki/Spazio_di_Baire" title="Spazio di Baire">Spazio di Baire</a><b> ·</b> <a href="/wiki/Spazio_topologico_noetheriano" title="Spazio topologico noetheriano">Spazio topologico noetheriano</a><b> ·</b> <a href="/wiki/Spazio_omogeneo" title="Spazio omogeneo">Spazio omogeneo</a><b> ·</b> <a href="/wiki/Orientazione" title="Orientazione">Orientazione</a></td></tr></tbody></table></td></tr><tr><th colspan="1" class="navbox_group" style="text-align:right;"><a href="/wiki/Topologia_differenziale" title="Topologia differenziale">Topologia differenziale</a></th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Variet%C3%A0_(geometria)" title="Varietà (geometria)">Varietà</a> (<a href="/wiki/Variet%C3%A0_differenziabile" title="Varietà differenziabile">differenziabile</a><b> ·</b> <a href="/wiki/Variet%C3%A0_parallelizzabile" title="Varietà parallelizzabile">parallelizzabile</a><b> ·</b> <a href="/wiki/3-variet%C3%A0" title="3-varietà">3-varietà</a><b> ·</b> <a href="/wiki/3-variet%C3%A0_irriducibile" title="3-varietà irriducibile">3-varietà irriducibile</a>)<b> ·</b> <a href="/wiki/Atlante_(topologia)" title="Atlante (topologia)">Atlante</a><b> ·</b> <a href="/wiki/Diffeomorfismo" title="Diffeomorfismo">Diffeomorfismo</a> (<a href="/wiki/Diffeomorfismo_locale" title="Diffeomorfismo locale">locale</a><b> ·</b> <a href="/wiki/Diffeomorfismo_di_Anosov" title="Diffeomorfismo di Anosov">di Anosov</a>)<b> ·</b> <a href="/wiki/Immersione_(geometria)" title="Immersione (geometria)">Immersione</a><b> ·</b> <a href="/wiki/Curva_(matematica)" title="Curva (matematica)">Curva</a><b> ·</b> <a href="/wiki/Superficie" title="Superficie">Superficie</a><b> ·</b> <a href="/wiki/Campo_vettoriale" title="Campo vettoriale">Campo vettoriale</a><b> ·</b> <a href="/wiki/Fibrato" title="Fibrato">Fibrato</a> (<a href="/wiki/Fibrato_principale" title="Fibrato principale">principale</a><b> ·</b> <a href="/wiki/Fibrato_vettoriale" title="Fibrato vettoriale">vettoriale</a><b> ·</b> <a href="/wiki/Variet%C3%A0_fibrata" title="Varietà fibrata">Varietà fibrata</a>)<b> ·</b> <a href="/wiki/Fibrato_tangente" title="Fibrato tangente">Fibrato tangente</a><b> ·</b> <a href="/wiki/Spazio_tangente" title="Spazio tangente">Spazio tangente</a><b> ·</b> <a href="/wiki/Fibrazione_di_Hopf" title="Fibrazione di Hopf">Fibrazione di Hopf</a><b> ·</b> <a href="/wiki/Variet%C3%A0_con_bordo" title="Varietà con bordo">Varietà con bordo</a><b> ·</b> <a href="/wiki/Teorema_dell%27intorno_tubolare" title="Teorema dell'intorno tubolare">Teorema dell'intorno tubolare</a><b> ·</b> <a href="/wiki/Somma_connessa" title="Somma connessa">Somma connessa</a><b> ·</b> <a href="/wiki/Teorema_di_Kneser-Milnor" title="Teorema di Kneser-Milnor">Teorema di Kneser-Milnor</a><b> ·</b> <a href="/wiki/Congettura_di_geometrizzazione_di_Thurston" title="Congettura di geometrizzazione di Thurston">Congettura di geometrizzazione di Thurston</a><b> ·</b> <a href="/wiki/Cobordismo" title="Cobordismo">Cobordismo</a><b> ·</b> <a href="/wiki/Dimensione_topologica" title="Dimensione topologica">Dimensione topologica</a><b> ·</b> <a href="/wiki/Topologia_in_dimensione_bassa" title="Topologia in dimensione bassa">Topologia in dimensione bassa</a><b> ·</b> <a href="/wiki/Chirurgia_di_Dehn" title="Chirurgia di Dehn">Chirurgia di Dehn</a><b> ·</b> <a href="/wiki/Trasversalit%C3%A0" title="Trasversalità">Trasversalità</a><b> ·</b> <a href="/wiki/Eversione_della_sfera" title="Eversione della sfera">Eversione della sfera</a><b> ·</b> <a href="/wiki/Teoria_delle_foliazioni" title="Teoria delle foliazioni">Teoria delle foliazioni</a><b> ·</b> <a href="/wiki/Decomposizione_JSJ" title="Decomposizione JSJ">Decomposizione JSJ</a></td></tr><tr><th colspan="1" class="navbox_group" style="text-align:right;"><a href="/wiki/Topologia_algebrica" title="Topologia algebrica">Topologia algebrica</a></th><td colspan="1" class="navbox_list navbox_odd"><table class="subnavbox"><tbody><tr><th class="subnavbox_group">Fondamenti</th><td colspan="1"><a href="/wiki/Spazio_semplicemente_connesso" title="Spazio semplicemente connesso">Spazio semplicemente connesso</a><b> ·</b> <a href="/wiki/Gruppo_fondamentale" title="Gruppo fondamentale">Gruppo fondamentale</a></td></tr><tr><th class="subnavbox_group">Omotopia</th><td colspan="1"><a href="/wiki/Arco_(topologia)" title="Arco (topologia)">Arco</a><b> ·</b> <a href="/wiki/Nerbo_(matematica)" title="Nerbo (matematica)">Nerbo</a><b> ·</b> <a href="/wiki/Omotopia" title="Omotopia">Omotopia</a><b> ·</b> <a href="/wiki/Gruppi_di_omotopia" title="Gruppi di omotopia">Gruppi di omotopia</a></td></tr><tr><th class="subnavbox_group">Omologia e coomologia</th><td colspan="1"><a href="/wiki/Omologia_(topologia)" title="Omologia (topologia)">Omologia</a><b> ·</b> <a href="/wiki/Omologia_singolare" title="Omologia singolare">Omologia singolare</a><b> ·</b> <a href="/wiki/Omologia_ciclica" title="Omologia ciclica">Omologia ciclica</a><b> ·</b> <a href="/wiki/Algebra_omologica" title="Algebra omologica">Algebra omologica</a><b> ·</b> <a href="/wiki/Coomologia_di_De_Rham" title="Coomologia di De Rham">Coomologia di De Rham</a><b> ·</b> <a href="/wiki/Categoria_abeliana" title="Categoria abeliana">Categoria abeliana</a></td></tr><tr><th class="subnavbox_group">Sollevamento</th><td colspan="1"><a href="/wiki/Sollevamento_(matematica)" title="Sollevamento (matematica)">Sollevamento</a><b> ·</b> <a href="/wiki/Teorema_del_sollevamento_dell%27omotopia" title="Teorema del sollevamento dell'omotopia">Teorema del sollevamento dell'omotopia</a><b> ·</b> <a href="/wiki/Teorema_di_unicit%C3%A0_del_sollevamento" title="Teorema di unicità del sollevamento">Teorema di unicità del sollevamento</a><b> ·</b> <a href="/wiki/Teorema_di_Van_Kampen" title="Teorema di Van Kampen">Teorema di Van Kampen</a></td></tr><tr><th class="subnavbox_group">Topologia algebrica avanzata</th><td colspan="1"><a href="/wiki/Grado_topologico" title="Grado topologico">Grado topologico</a><b> ·</b> <a href="/wiki/Indice_di_avvolgimento" title="Indice di avvolgimento">Indice di avvolgimento</a><b> ·</b> <a href="/wiki/Indice_di_un_campo_vettoriale" title="Indice di un campo vettoriale">Indice di un campo vettoriale</a><b> ·</b> <a href="/wiki/Rivestimento_(topologia)" title="Rivestimento (topologia)">Rivestimento</a><b> ·</b> <a href="/wiki/Numero_di_Betti" title="Numero di Betti">Numero di Betti</a><b> ·</b> <a href="/wiki/Successione_di_Mayer-Vietoris" title="Successione di Mayer-Vietoris">Successione di Mayer-Vietoris</a><b> ·</b> <a href="/wiki/Successione_esatta" title="Successione esatta">Successione esatta</a><b> ·</b> <a href="/wiki/Successione_spettrale" title="Successione spettrale">Successione spettrale</a><b> ·</b> <a href="/wiki/Complesso_simpliciale" title="Complesso simpliciale">Complesso simpliciale</a><b> ·</b> <a href="/wiki/Complesso_di_celle" title="Complesso di celle">Complesso di celle</a><b> ·</b> <a href="/wiki/Complesso_di_catene" title="Complesso di catene">Complesso di catene</a><b> ·</b> <a href="/wiki/Schema_simpliciale" title="Schema simpliciale">Schema simpliciale</a></td></tr><tr><th class="subnavbox_group">Superfici</th><td colspan="1"><a href="/wiki/Caratteristica_di_Eulero" title="Caratteristica di Eulero">Caratteristica di Eulero</a><b> ·</b> <a href="/wiki/Formula_di_Eulero_per_i_poliedri" title="Formula di Eulero per i poliedri">Formula di Eulero per i poliedri</a><b> ·</b> <a href="/wiki/Genere_(matematica)" title="Genere (matematica)">Genere</a><b> ·</b> <a href="/wiki/Taglio_(topologia)" title="Taglio (topologia)">Taglio</a><b> ·</b> <a href="/wiki/Superficie_incompressibile" title="Superficie incompressibile">Superficie incompressibile</a><b> ·</b> <a href="/wiki/Classificazione_delle_superfici" title="Classificazione delle superfici">Classificazione delle superfici</a><b> ·</b> <a href="/wiki/Mapping_class_group" title="Mapping class group">Mapping class group</a><b> ·</b> <a href="/wiki/Teorema_della_palla_pelosa" title="Teorema della palla pelosa">Teorema della palla pelosa</a><b> ·</b> <a href="/wiki/Teorema_di_Poincar%C3%A9-Hopf" title="Teorema di Poincaré-Hopf">Teorema di Poincaré-Hopf</a><b> ·</b> <a href="/wiki/Congettura_di_Poincar%C3%A9" title="Congettura di Poincaré">Congettura di Poincaré</a><b> ·</b> <a href="/wiki/Congettura_di_Hodge" title="Congettura di Hodge">Congettura di Hodge</a></td></tr></tbody></table></td></tr><tr><th colspan="1" class="navbox_group" style="text-align:right;">Topologi di rilievo</th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a><b> ·</b> <a href="/wiki/Felix_Hausdorff" title="Felix Hausdorff">Felix Hausdorff</a><b> ·</b> <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a><b> ·</b> <a href="/wiki/Eduard_%C4%8Cech" title="Eduard Čech">Eduard Čech</a><b> ·</b> <a href="/wiki/John_Milnor" title="John Milnor">John Milnor</a><b> ·</b> <a href="/w/index.php?title=Pierre_Samuel&action=edit&redlink=1" class="new" title="Pierre Samuel (la pagina non esiste)">Pierre Samuel</a><b> ·</b> <a href="/w/index.php?title=Norman_Steenrod&action=edit&redlink=1" class="new" title="Norman Steenrod (la pagina non esiste)">Norman Steenrod</a><b> ·</b> <a href="/wiki/Ren%C3%A9_Thom" title="René Thom">René Thom</a><b> ·</b> <a href="/wiki/Samuel_Eilenberg" title="Samuel Eilenberg">Samuel Eilenberg</a><b> ·</b> <a href="/wiki/Andrej_Nikolaevi%C4%8D_Kolmogorov" title="Andrej Nikolaevič Kolmogorov">Andrej Nikolaevič Kolmogorov</a><b> ·</b> <a href="/wiki/Stephen_Smale" title="Stephen Smale">Stephen Smale</a><b> ·</b> <a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Michael Atiyah</a><b> ·</b> <a href="/wiki/William_Thurston" title="William Thurston">William Thurston</a><b> ·</b> <a href="/wiki/Marston_Morse" title="Marston Morse">Marston Morse</a><b> ·</b> <a href="/wiki/Luitzen_Brouwer" title="Luitzen Brouwer">Luitzen Brouwer</a></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r140554510">.mw-parser-output .CdA{border:1px solid #aaa;width:100%;margin:auto;font-size:90%;padding:2px}.mw-parser-output .CdA 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title="tedesco">DE</abbr></span>) <a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4316404-3">4316404-3</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_della_Dieta_nazionale_del_Giappone" title="Biblioteca della Dieta nazionale del Giappone">NDL</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr>, <abbr title="giapponese">JA</abbr></span>) <a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00567231">00567231</a></span></td></tr></tbody></table> <div class="noprint" style="width:100%; padding: 3px 0; display: flex; flex-wrap: wrap; row-gap: 4px; column-gap: 8px; box-sizing: border-box;"><div style="flex-grow: 1"><style data-mw-deduplicate="TemplateStyles:r140555418">.mw-parser-output .itwiki-template-occhiello{width:100%;line-height:25px;border:1px solid #CCF;background-color:#F0EEFF;box-sizing:border-box}.mw-parser-output .itwiki-template-occhiello-progetto{background-color:#FAFAFA}@media screen{html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}</style><div class="itwiki-template-occhiello"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Crystal128-kmplot.svg" class="mw-file-description" title="Matematica"><img alt=" " src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/25px-Crystal128-kmplot.svg.png" decoding="async" width="25" height="25" class="mw-file-element" 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