{"id":29349,"date":"2024-10-22T13:00:08","date_gmt":"2024-10-22T13:00:08","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=29349"},"modified":"2024-10-23T14:38:12","modified_gmt":"2024-10-23T14:38:12","slug":"limite-a-linfini-definitions-et-exemples","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/fr\/limite-a-linfini-definitions-et-exemples\/","title":{"rendered":"Limite \u00e0 l&#8217;infini : D\u00e9finitions et exemples"},"content":{"rendered":"<p><center><\/p>\n<h1>Limite \u00e0 l&#8217;infini : D\u00e9finitions et exemples<\/h1>\n<p><em><\/p>\n<p><strong>R\u00e9sum\u00e9 :<\/strong><br \/>\nDans ce cours, nous aborderons les limites \u00e0 l&#8217;infini, d\u00e9crivant le comportement de la fonction <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> lorsque <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> tend vers l&#8217;infini. Nous expliquerons des limites de base telles que <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x\\to \\infty} \\frac{1}{x} = 0<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x\\to \\infty} k = k<\/span>, ainsi que des propri\u00e9t\u00e9s alg\u00e9briques similaires \u00e0 celles des limites finies.\n<\/p>\n<p><\/em><\/p>\n<p><strong>Objectifs d&#8217;apprentissage :<\/strong><br \/>\n\u00c0 la fin de ce cours, l&#8217;\u00e9tudiant sera capable de :<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>D\u00e9crire<\/strong> le comportement de <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> lorsque <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> tend vers l&#8217;infini.<\/li>\n<li><strong>D\u00e9finir<\/strong> la limite \u00e0 l&#8217;infini en utilisant une notation math\u00e9matique formelle.<\/li>\n<li><strong>Appliquer<\/strong> les propri\u00e9t\u00e9s alg\u00e9briques dans le calcul des limites \u00e0 l&#8217;infini.<\/li>\n<li><strong>Distinguer<\/strong> entre diff\u00e9rents cas de limites des fonctions rationnelles \u00e0 l&#8217;infini.<\/li>\n<li><strong>D\u00e9montrer<\/strong> la validit\u00e9 des propri\u00e9t\u00e9s de somme, de soustraction, de multiplication, de division et de puissances des limites \u00e0 l&#8217;infini.<\/li>\n<li><strong>R\u00e9soudre<\/strong> des exercices pratiques sur les limites \u00e0 l&#8217;infini dans diff\u00e9rentes fonctions.<\/li>\n<\/ol>\n<p><u>Table des mati\u00e8res :<\/u><br \/>\n<a href=\"#1\">Introduction<\/a><br \/>\n<a href=\"#2\">D\u00e9finition de la limite \u00e0 l&#8217;infini<\/a><br \/>\n<a href=\"#3\">Limites de base \u00e0 l&#8217;infini<\/a><br \/>\n<a href=\"#4\">Alg\u00e8bre des limites \u00e0 l&#8217;infini<\/a><br \/>\n<a href=\"#5\">Limite \u00e0 l&#8217;infini dans les fonctions rationnelles<\/a><br \/>\n<a href=\"#6\">Exemples de limites \u00e0 l&#8217;infini<\/a><br \/>\n<\/center><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/MjjSAQLeNBE\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Introduction<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=41s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">L&#8217;un des \u00e9l\u00e9ments les plus caract\u00e9ristiques du calcul est l&#8217;infini et la limite \u00e0 l&#8217;infini.<\/span><\/strong><\/a> Le concept d&#8217;infini ne d\u00e9signe pas un nombre r\u00e9el, mais plut\u00f4t une grandeur qui d\u00e9passe toute borne r\u00e9elle. Par exemple, pour la fonction <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = 1\/x<\/span>, si l&#8217;on se demande quel est son comportement lorsque <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> devient aussi grand que possible, c&#8217;est-\u00e0-dire lorsque <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> tend vers l&#8217;infini (<span class=\"katex-eq\" data-katex-display=\"false\">x\\to \\infty<\/span>), on observe que <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> peut se rapprocher de z\u00e9ro aussi pr\u00e8s que l&#8217;on veut. Dans ce cas, nous \u00e9crivons :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to + \\infty}\\dfrac{1}{x} = 0<\/span>\n<p style=\"text-align: justify; color: #000000;\">Graphiquement, cela ressemble \u00e0 ceci :<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-fX0lg2ICTTU\/YGD0hNJWI6I\/AAAAAAAAEwQ\/0v8hW6ARkQYDIzT_eG5WVgZ0-pPwwPBwgCLcBGAsYHQ\/s0\/limiteAlinfinito.PNG\" alt=\"limite \u00e0 l'infini\" class=\"alignnone size-full lazyload\" width=\"400\" height=\"300\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-fX0lg2ICTTU\/YGD0hNJWI6I\/AAAAAAAAEwQ\/0v8hW6ARkQYDIzT_eG5WVgZ0-pPwwPBwgCLcBGAsYHQ\/s0\/limiteAlinfinito.PNG\" alt=\"limite \u00e0 l'infini\" class=\"alignnone size-full lazyload\" width=\"400\" height=\"300\" \/><\/noscript><\/center><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>D\u00e9finition de la limite \u00e0 l&#8217;infini<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=144s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u00c0 partir de cette id\u00e9e que nous venons d&#8217;introduire<\/span><\/strong><\/a>, nous pouvons formuler la d\u00e9finition math\u00e9matique de la limite \u00e0 l&#8217;infini :<\/p>\n<p style=\"text-align: justify; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}f(x) = L := (\\forall\\epsilon\\gt 0) (\\exists M\\in\\mathbb{R})(M\\lt x \\rightarrow |f(x) - L|\\lt \\epsilon )<\/span>\n<p style=\"text-align: justify; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to -\\infty}f(x) = L := (\\forall\\epsilon\\gt 0) (\\exists N\\in\\mathbb{R})(x\\lt N \\rightarrow |f(x) - L|\\lt \\epsilon )<\/span>\n<p style=\"text-align: justify; color: #000000;\">L&#8217;id\u00e9e intuitive de cette limite nous indique ce qui se passe avec <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> lorsque <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> s&#8217;\u00e9loigne autant que l&#8217;on veut de l&#8217;origine, soit vers la droite, soit vers la gauche. La strat\u00e9gie pour le calcul des limites \u00e0 l&#8217;infini n&#8217;est pas tr\u00e8s diff\u00e9rente de celle des limites finies, car son alg\u00e8bre est pratiquement la m\u00eame, \u00e0 condition de prendre en compte les r\u00e9sultats suivants :<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Limites de base \u00e0 l&#8217;infini<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=450s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u00c0 partir de ces d\u00e9finitions, nous pouvons d\u00e9montrer<\/span><\/strong><\/a> les limites de base suivantes :<\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}k = k <\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}\\dfrac{1}{x} = 0 <\/span><\/li>\n<\/ol>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000080;\">D\u00e9monstration :<\/span><\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li>Selon la d\u00e9finition de la limite \u00e0 l&#8217;infini, <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}k = k <\/span> est \u00e9quivalent \u00e0 : <span class=\"katex-eq\" data-katex-display=\"false\">(\\forall\\epsilon\\gt 0) (\\exists M\\in\\mathbb{R})\\left(M\\lt x \\rightarrow \\left|k-k\\right|\\lt \\epsilon \\right)<\/span>. Comme <span class=\"katex-eq\" data-katex-display=\"false\">\\left|k-k\\right|=0\\lt \\epsilon<\/span> est toujours vrai pour tout <span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon \\gt 0<\/span> et pour toute valeur de <span class=\"katex-eq\" data-katex-display=\"false\">M<\/span>, la limite est v\u00e9rifi\u00e9e.\n<p>&nbsp;<\/li>\n<li>D&#8217;apr\u00e8s la d\u00e9finition <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}k = k <\/span>, cela signifie : <span class=\"katex-eq\" data-katex-display=\"false\">(\\forall\\epsilon\\gt 0) (\\exists M\\in\\mathbb{R})\\left(M\\lt x \\rightarrow \\left|\\dfrac{1}{x}\\right|\\lt \\epsilon \\right)<\/span>. Cette implication est satisfaite imm\u00e9diatement si nous prenons <span class=\"katex-eq\" data-katex-display=\"false\">M=1\/\\epsilon<\/span>, de sorte que la limite est v\u00e9rifi\u00e9e.\n<p>&nbsp;<\/li>\n<\/ol>\n<p style=\"text-align: justify; color: #000000;\">Ces d\u00e9monstrations sont \u00e9galement applicables lorsque <span class=\"katex-eq\" data-katex-display=\"false\">x\\to+\\infty<\/span>.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Alg\u00e8bre des limites \u00e0 l&#8217;infini<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=620s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">L&#8217;alg\u00e8bre des limites \u00e0 l&#8217;infini est analogue \u00e0 celle des limites finies.<\/span><\/strong><\/a> Si <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm \\infty}f(x) = L<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm \\infty}g(x) = M<\/span>, alors les r\u00e8gles suivantes s&#8217;appliquent :<\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><strong>Somme et soustraction des limites :<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}(f(x)\\pm g(x)) = L \\pm M<\/span><\/li>\n<li><strong>Multiplication par une constante :<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}cf(x) = cL<\/span><\/li>\n<li><strong>Produit des limites :<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}f(x)g(x) = LM<\/span><\/li>\n<li><strong>Division des limites :<\/strong> \u00c0 condition que <span class=\"katex-eq\" data-katex-display=\"false\">M\\neq 0<\/span>, alors <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}f(x)\/g(x)=L\/M<\/span><\/li>\n<li><strong>Puissances des limites :<\/strong> Si <span class=\"katex-eq\" data-katex-display=\"false\">p,q \\in\\mathbb{Z}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">q\\neq 0<\/span>, alors <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}[f(x)]^{p\/q} = L^{p\/q}<\/span>. Si <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> est pair, on suppose que <span class=\"katex-eq\" data-katex-display=\"false\">L\\geq 0<\/span>.<\/li>\n<\/ol>\n<p style=\"text-align: justify; color: #000000;\">En r\u00e9alit\u00e9, la d\u00e9monstration de toutes ces propri\u00e9t\u00e9s est analogue \u00e0 celle des <a href=\"https:\/\/toposuranos.com\/la-definicion-de-limite-demostraciones-y-teoremas\/\" target=\"_blank\" rel=\"noopener\">limites finies<\/a>.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Limite \u00e0 l&#8217;infini dans les fonctions rationnelles<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=792s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Une fonction rationnelle est celle qui peut \u00eatre exprim\u00e9e comme le quotient de deux polyn\u00f4mes.<\/span><\/strong><\/a> Lors du calcul des limites \u00e0 l&#8217;infini de ce type de fonctions, on peut observer une propri\u00e9t\u00e9 tr\u00e8s utile :<\/p>\n<p style=\"text-align: justify; color: #000000;\">Supposons que nous voulons calculer <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\infty}P(x)\/Q(x)<\/span> :<\/p>\n<ul style=\"text-align: justify; color: #000000;\">\n<li>Si le degr\u00e9 de <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> est sup\u00e9rieur \u00e0 celui de <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span>, alors la fonction <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> cro\u00eetra sans limite lorsque <span class=\"katex-eq\" data-katex-display=\"false\">x\\to\\infty<\/span> (la limite n&#8217;existera pas).<\/li>\n<li>Si le degr\u00e9 de <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> est inf\u00e9rieur \u00e0 celui de <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span>, alors la limite sera z\u00e9ro.<\/li>\n<li>Enfin, si le degr\u00e9 de <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> est \u00e9gal \u00e0 celui de <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span>, alors la limite sera \u00e9gale au rapport des coefficients des termes de plus haut degr\u00e9.<\/li>\n<\/ul>\n<p style=\"text-align: justify; color: #000000;\">La meilleure chose \u00e0 propos de ce r\u00e9sultat est que, comme nous le verrons dans les exemples suivants, il fonctionne de mani\u00e8re analogue, m\u00eame si les puissances impliqu\u00e9es ne sont pas des entiers.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h2>Exemples de limites \u00e0 l&#8217;infini<\/h2>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{x+1}{x^2+3}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=907s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to -\\infty}\\dfrac{2x^3 + 7}{x^3 - x^2 + x + 7}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=986s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{9x^4 + x}{2x^4 + 5x^2 - x + 6}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1049s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{10x^5 + x4 + 31}{x^4 - 7x^3 + 7x^2 + 9}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1111s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{2\\sqrt{x}+x^{-1}}{3x - 7}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1220s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to -\\infty}\\dfrac{2x^{5\/3} - x^{1\/3} + 7}{x^{8\/5}+3x + \\sqrt{x}}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1284s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{\\sqrt[3]{x}-5x+3}{2x + x^{2\/3} - 4}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1406s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{x^{8\/3}+2x + \\sqrt{x}}{x^2+x-3}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1521s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Limite \u00e0 l&#8217;infini : D\u00e9finitions et exemples R\u00e9sum\u00e9 : Dans ce cours, nous aborderons les limites \u00e0 l&#8217;infini, d\u00e9crivant le comportement de la fonction lorsque tend vers l&#8217;infini. Nous expliquerons des limites de base telles que et , ainsi que des propri\u00e9t\u00e9s alg\u00e9briques similaires \u00e0 celles des limites finies. Objectifs d&#8217;apprentissage : \u00c0 la fin [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":29336,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":118,"footnotes":""},"categories":[866,569],"tags":[],"class_list":["post-29349","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 v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Limite \u00e0 l&#039;infini : D\u00e9finitions et exemples - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Ne pas comprendre les limites \u00e0 l&#039;infini vous co\u00fbte des points ! 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