{"id":29471,"date":"2024-11-12T13:00:20","date_gmt":"2024-11-12T13:00:20","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=29471"},"modified":"2024-11-13T07:42:26","modified_gmt":"2024-11-13T07:42:26","slug":"limites-infinies-et-divergence","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/fr\/limites-infinies-et-divergence\/","title":{"rendered":"Limites infinies et divergence"},"content":{"rendered":"<style>\np {\ntext-align: justify;}\n<\/style>\n<p><center><\/p>\n<h1>Limites infinies et divergence<\/h1>\n<p style=\"text-align:center;\"><em><strong>R\u00e9sum\u00e9 :<\/strong><br \/>\nDans ce cours, nous aborderons les limites infinies et les diff\u00e9rents types de divergence dans les limites, en explorant des concepts fondamentaux pour comprendre pourquoi certaines fonctions ne convergent pas vers une valeur r\u00e9elle d\u00e9finie. Nous passerons en revue les limites lat\u00e9rales distinctes, les fonctions oscillant \u00e0 l\u2019infini et les situations o\u00f9 les limites n\u2019existent pas en raison de probl\u00e8mes de domaine ou de croissance illimit\u00e9e.<br \/>\n<\/em><\/p>\n<p style=\"text-align:center;\"><strong>Objectifs d\u2019apprentissage :<\/strong><br \/>\n\u00c0 la fin de ce cours, l\u2019\u00e9tudiant sera capable de\n<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>D\u00e9finir<\/strong> les limites divergentes et reconna\u00eetre quand une limite est divergente.<\/li>\n<li><strong>Identifier<\/strong> les diff\u00e9rents types de divergence dans les limites, comme les limites lat\u00e9rales distinctes et les limites infinies.<\/li>\n<li><strong>Analyser<\/strong> les situations o\u00f9 une fonction pose des probl\u00e8mes de domaine et comment cela affecte l\u2019existence de la limite.<\/li>\n<li><strong>\u00c9valuer<\/strong> les limites lat\u00e9rales pour d\u00e9terminer si elles sont distinctes et l\u2019impact sur la convergence de la limite.<\/li>\n<li><strong>Calculer<\/strong> les limites infinies et diff\u00e9rencier les limites qui divergent vers l\u2019infini positif et n\u00e9gatif.<\/li>\n<\/ol>\n<p style=\"text-align:center;\">\n<u><strong>TABLE DES MATI\u00c8RES<\/strong><\/u> :<br \/>\n<a href=\"#1\"><strong>Quand dit-on qu\u2019une limite est divergente ?<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>Types de divergence dans les limites<\/strong><\/a><br \/>\n<a href=\"#3\">Limites avec des probl\u00e8mes de domaine<\/a><br \/>\n<a href=\"#4\">Limites lat\u00e9rales distinctes<\/a><br \/>\n<a href=\"#5\">Limite des fonctions oscillant \u00e0 l\u2019infini<\/a><br \/>\n<a href=\"#6\">Limites infinies<\/a><br \/>\n<a href=\"#7\">Limites infinies \u00e0 l\u2019infini<\/a>\n<\/p>\n<p><\/center><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/SFBMSd0Q7Io\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p>Cette fois, nous ne nous limiterons pas aux <strong>limites infinies<\/strong> mais aborderons \u00e9galement les <strong>limites divergentes<\/strong> en g\u00e9n\u00e9ral. Les limites divergentes nous renseignent sur la mani\u00e8re dont une fonction semble ne pas converger, ce qui peut se produire de diverses fa\u00e7ons.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Quand dit-on qu\u2019une limite est divergente ?<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=SFBMSd0Q7Io&amp;t=106s\" target=\"_blank\" rel=\"noopener\"><strong>On dit qu\u2019une limite est divergente lorsqu\u2019elle ne converge pas vers une valeur r\u00e9elle.<\/strong><\/a> Ce qui peut sembler \u00e9vident peut en fait se produire de diff\u00e9rentes mani\u00e8res :<\/p>\n<ul>\n<li>Lorsque les limites lat\u00e9rales sont distinctes ou inexistantes, les limites bilat\u00e9rales n\u2019existent pas.<\/li>\n<li>Si la fonction n\u2019est pas bien d\u00e9finie, cro\u00eet de fa\u00e7on illimit\u00e9e ou oscille \u00e0 l\u2019infini en s\u2019approchant du point o\u00f9 la limite est calcul\u00e9e, alors la limite lat\u00e9rale ne peut pas exister.<\/li>\n<\/ul>\n<p>Cela peut s\u2019appliquer, avec certaines particularit\u00e9s, aux limites finies comme aux limites \u00e0 l\u2019infini, et selon le cas, un type de divergence se produira.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Types de divergence dans les limites<\/h2>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Limites avec des probl\u00e8mes de domaine<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=SFBMSd0Q7Io&amp;t=210s\" target=\"_blank\" rel=\"noopener\"><strong>Lorsque nous essayons de calculer une limite de la forme<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x\\to x_0}f(x)<\/span> ou <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x\\to +\\infty}f(x),<\/span> nous nous attendons \u00e0 ce que <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> soit au moins bien d\u00e9finie pour des valeurs proches de <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> ou pour un intervalle de la forme <span class=\"katex-eq\" data-katex-display=\"false\">[a,+\\infty[, <\/span> respectivement. Si cela n\u2019est pas le cas, aucune des deux d\u00e9finitions des limites ne pourrait avoir de sens ; la fonction ne peut \u00abtendre\u00bb vers une valeur si elle n\u2019est pas d\u00e9finie. Dans ces cas, on \u00e9crit simplement que la limite n\u2019existe pas : <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x\\to x_0}f(x)=\\cancel{\\exists}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x\\to +\\infty}f(x)=\\cancel{\\exists},<\/span> selon le cas. Il en est de m\u00eame pour les limites lat\u00e9rales, et il n\u2019y a rien de plus \u00e0 ajouter \u00e0 ce type de situation.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Limites lat\u00e9rales distinctes<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=SFBMSd0Q7Io&amp;t=180s\" target=\"_blank\" rel=\"noopener\"><strong>Consid\u00e9rons une fonction de la forme<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = x\/|x|<\/span> et calculons la limite lorsque <span class=\"katex-eq\" data-katex-display=\"false\">x\\to 0<\/span>. La premi\u00e8re chose que nous remarquerons est que<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0^+} f(x) = 1<\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0^-} f(x) = -1<\/span>\n<p>Dans ce cas, nous constatons que bien que les limites lat\u00e9rales existent, elles sont distinctes. Lorsque cela se produit, nous disons simplement que la limite (bilat\u00e9rale) ne converge pas, et donc :<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0} f(x) = \\cancel{\\exists}<\/span>\n<p><a name=\"5\"><\/a><\/p>\n<h3>Limite des fonctions oscillant \u00e0 l\u2019infini<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=SFBMSd0Q7Io&amp;t=415s\" target=\"_blank\" rel=\"noopener\"><strong>Il existe \u00e9galement des fonctions qui, au lieu de tendre vers une certaine valeur,<\/strong><\/a> commencent \u00e0 osciller dans une certaine plage. Un exemple de cela pourrait \u00eatre une fonction de la forme <span class=\"katex-eq\" data-katex-display=\"false\">f(x)= \\sin(1\/x)<\/span>. Si nous observons ce qui se passe avec cette fonction lorsque <span class=\"katex-eq\" data-katex-display=\"false\">x\\to 0<\/span>, nous verrons qu\u2019elle oscille \u00e0 l\u2019infini.<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-vrdcIAmLV50\/YGoC903Q62I\/AAAAAAAAE1M\/6AM7D-RM9lAy9ZrG105MbfCN8ltu6J09ACLcBGAsYHQ\/s0\/sin1sobrex.PNG\" alt=\"f(x) = sin(1\/x)\" class=\"alignnone size-full lazyload\" width=\"856\" height=\"442\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-vrdcIAmLV50\/YGoC903Q62I\/AAAAAAAAE1M\/6AM7D-RM9lAy9ZrG105MbfCN8ltu6J09ACLcBGAsYHQ\/s0\/sin1sobrex.PNG\" alt=\"f(x) = sin(1\/x)\" class=\"alignnone size-full lazyload\" width=\"856\" height=\"442\" \/><\/noscript><\/p>\n<p>Lorsque des situations similaires se produisent, nous disons simplement que la limite n\u2019existe pas.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h3>Limites infinies<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=SFBMSd0Q7Io&amp;t=658s\" target=\"_blank\" rel=\"noopener\"><strong>Voyons ce qui se passe avec la fonction<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = 1\/x.<\/span> La premi\u00e8re chose que nous verrons est que lorsque <span class=\"katex-eq\" data-katex-display=\"false\">x\\to 0<\/span>, la valeur de <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> cro\u00eet de fa\u00e7on illimit\u00e9e, mais la fa\u00e7on dont elle le fera d\u00e9pendra du c\u00f4t\u00e9 o\u00f9 la limite est calcul\u00e9e. Intuitivement, nous \u00e9crirons<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0^+} \\dfrac{1}{x} = +\\infty<\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0^-} \\dfrac{1}{x} = -\\infty<\/span>\n<p>Avec cette notation, nous ne disons pas que la limite existe d\u2019une certaine mani\u00e8re ; nous indiquons simplement comment cette limite n\u2019existe pas. Contrairement aux cas pr\u00e9c\u00e9dents o\u00f9 la limite n\u2019existe pas et ne converge pas vers une valeur concr\u00e8te ; dans ce cas, elle diverge parce que sa grandeur d\u00e9passe tout nombre r\u00e9el.<\/p>\n<p>Ce que nous venons de revoir peut \u00eatre formalis\u00e9 par les d\u00e9finitions suivantes :<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to x_0^+}f(x) = +\\infty := \\left(\\forall M \\in \\mathbb{R}\\right)\\left( \\exists \\delta \\gt 0 \\right) ( x_0 \\lt x \\lt x_0 + \\delta \\rightarrow M \\lt f(x) )<\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to x_0^-}f(x) = +\\infty := \\left(\\forall M \\in \\mathbb{R}\\right)\\left( \\exists \\delta \\gt 0 \\right) ( x_0 - \\delta \\lt x \\lt x_0 \\rightarrow M \\lt f(x) )<\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to x_0}f(x) = +\\infty := \\left(\\lim_{x\\to x_0^+}f(x) = +\\infty \\right) \\wedge \\left(\\lim_{x\\to x_0^-}f(x) = +\\infty \\right)<\/span>\n<p>Et de mani\u00e8re analogue :<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to x_0^+}f(x) = -\\infty := \\left(\\forall m \\in \\mathbb{R}\\right)\\left( \\exists \\delta \\gt 0 \\right) ( x_0 \\lt x \\lt x_0 + \\delta \\rightarrow f(x) \\lt m )<\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to x_0^-}f(x) = -\\infty := \\left(\\forall m \\in \\mathbb{R}\\right)\\left( \\exists \\delta \\gt 0 \\right) ( x_0 - \\delta \\lt x \\lt x_0 \\rightarrow f(x) \\lt m )<\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to x_0}f(x) = -\\infty := \\left(\\lim_{x\\to x_0^+}f(x) = -\\infty \\right) \\wedge \\left(\\lim_{x\\to x_0^-}f(x) = -\\infty \\right)<\/span>\n<p>Parfois, on parle \u00e9galement de limite tendant vers l\u2019infini (sans signe)<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to x_0}f(x) = \\infty := \\lim_{x\\to x_0}|f(x)| = +\\infty <\/span>\n<p><a name=\"7\"><\/a><\/p>\n<h3>Limites infinies \u00e0 l\u2019infini<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=SFBMSd0Q7Io&amp;t=1147s\" target=\"_blank\" rel=\"noopener\"><strong>De mani\u00e8re similaire aux limites pr\u00e9c\u00e9dentes,<\/strong><\/a> il est possible de d\u00e9finir les limites infinies \u00e0 l\u2019infini. Par exemple :<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to +\\infty}f(x) = +\\infty := \\left(\\forall M \\in \\mathbb{R}\\right)\\left( \\exists N \\in\\mathbb{R} \\right) ( N\\lt x \\rightarrow M \\lt f(x) )<\/span>\n<p>Avec cela, nous avons vu toutes les fa\u00e7ons dont les limites des fonctions peuvent diverger.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Limites infinies et divergence R\u00e9sum\u00e9 : Dans ce cours, nous aborderons les limites infinies et les diff\u00e9rents types de divergence dans les limites, en explorant des concepts fondamentaux pour comprendre pourquoi certaines fonctions ne convergent pas vers une valeur r\u00e9elle d\u00e9finie. Nous passerons en revue les limites lat\u00e9rales distinctes, les fonctions oscillant \u00e0 l\u2019infini et [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":29458,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":15,"footnotes":""},"categories":[866,569],"tags":[],"class_list":["post-29471","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-calcul-differentiel","category-mathematiques"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Limites infinies et divergence - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Ici, nous allons revoir comment sont d\u00e9finies les limites infinies et ce qu\u2019est la divergence des fonctions. 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