{"id":29572,"date":"2024-11-18T17:34:03","date_gmt":"2024-11-18T17:34:03","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=29572"},"modified":"2024-11-18T17:34:03","modified_gmt":"2024-11-18T17:34:03","slug":"limite-et-continuite","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/fr\/limite-et-continuite\/","title":{"rendered":"Limite et Continuit\u00e9"},"content":{"rendered":"<style>\np {\ntext-align:justify;\n}\n<\/style>\n<p><center><\/p>\n<h1>Limite et Continuit\u00e9<\/h1>\n<p><\/center><\/p>\n<p style=\"text-align:center;\"><em><strong>R\u00e9sum\u00e9 :<\/strong><\/em><br \/>\nCe cours traite de la relation entre la limite et la continuit\u00e9 d&#8217;une fonction, en commen\u00e7ant par une explication intuitive et formelle du terme. La continuit\u00e9 en un point et sur un ensemble est explor\u00e9e, en d\u00e9taillant les conditions n\u00e9cessaires pour qu&#8217;une fonction soit continue. Les propri\u00e9t\u00e9s alg\u00e9briques des fonctions continues sont \u00e9galement pr\u00e9sent\u00e9es, y compris les sommes, produits, quotients et compositions de fonctions. Enfin, des exemples pratiques sont r\u00e9solus pour analyser la continuit\u00e9 de diff\u00e9rentes fonctions et calculer des limites, en les reliant au concept de continuit\u00e9.\n<\/p>\n<p style=\"text-align:center;\"><strong>Objectifs d&#8217;apprentissage :<\/strong><br \/>\n\u00c0 la fin de ce cours, l&#8217;\u00e9tudiant sera capable de :<\/p>\n<ol>\n<li><strong>Analyser<\/strong> la continuit\u00e9 d&#8217;une fonction en un point en v\u00e9rifiant les conditions d\u00e9finies.<\/li>\n<li><strong>\u00c9valuer<\/strong> si une fonction est continue sur un ensemble donn\u00e9 en utilisant la d\u00e9finition formelle de la continuit\u00e9.<\/li>\n<li><strong>D\u00e9montrer<\/strong> la continuit\u00e9 d&#8217;une fonction \u00e0 partir de ses propri\u00e9t\u00e9s alg\u00e9briques.<\/li>\n<li><strong>Classer<\/strong> les points de discontinuit\u00e9 en diff\u00e9rents types selon les conditions qui ne sont pas remplies.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>TABLE DES MATI\u00c8RES :<\/u><\/strong><br \/>\n<a href=\"#1\">Introduction<\/a><br \/>\n<a href=\"#2\">Continuit\u00e9 en un Point<\/a><br \/>\n<a href=\"#3\">Continuit\u00e9 sur un Ensemble<\/a><br \/>\n<a href=\"#4\">Propri\u00e9t\u00e9s des Fonctions Continues<\/a><br \/>\n<a href=\"#5\">Exercices<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/4kehfJcyPRg\" 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 ><a href=\"https:\/\/www.youtube.com\/watch?v=4kehfJcyPRg&amp;t=109s\" target=\"_blank\" rel=\"noopener\"><strong>De mani\u00e8re intuitive, la continuit\u00e9<\/strong><\/a> d&#8217;une fonction est comprise comme la propri\u00e9t\u00e9 qui permet de tracer son graphique \u00absans lever le crayon\u00bb. Bien que cette notion soit assez simple \u00e0 comprendre, elle n&#8217;est pas suffisante si l&#8217;on souhaite obtenir la rigueur du raisonnement math\u00e9matique. Un minimum de formalit\u00e9s est n\u00e9cessaire. D&#8217;un point de vue formel, la continuit\u00e9 des fonctions est \u00e9troitement li\u00e9e au concept de limite, que nous \u00e9tudierons en d\u00e9tail.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h3>Continuit\u00e9 en un Point<\/h3>\n<p ><a href=\"https:\/\/www.youtube.com\/watch?v=4kehfJcyPRg&amp;t=152s\" target=\"_blank\" rel=\"noopener\"><strong>Une fonction <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> est continue en un point<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">x=x_0<\/span> si, et seulement si, les conditions suivantes sont remplies :<\/p>\n<ul >\n<li><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> est d\u00e9finie en <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/li>\n<li>La limite <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} f(x)<\/span> existe<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} f(x) = f(x_0)<\/span><\/li>\n<\/ul>\n<p >Tout cela peut \u00eatre r\u00e9sum\u00e9 math\u00e9matiquement par l&#8217;expression suivante :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\forall \\epsilon \\gt 0 \\right) \\left(\\exists \\delta \\gt 0 \\right) \\left(0\\lt |x-x_0|\\lt \\delta \\rightarrow |f(x) - f(x_0)|\\lt \\epsilon\\right)<\/span>\n<p >Ceci est pr\u00e9cis\u00e9ment la d\u00e9finition math\u00e9matique de la limite, o\u00f9 la valeur \u00abL\u00bb est remplac\u00e9e par \u00abf(<span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span>)\u00bb<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Continuit\u00e9 sur un Ensemble<\/h3>\n<p ><a href=\"https:\/\/www.youtube.com\/watch?v=4kehfJcyPRg&amp;t=462s\" target=\"_blank\" rel=\"noopener\"><strong>La d\u00e9finition de la continuit\u00e9<\/strong><\/a> en un point peut \u00eatre \u00e9tendue \u00e0 la continuit\u00e9 sur tous les points d&#8217;un certain ensemble. Nous disons que <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> est continue sur un ensemble <span class=\"katex-eq\" data-katex-display=\"false\">I\\subseteq Dom(f)<\/span> si elle est continue pour tous les points <span class=\"katex-eq\" data-katex-display=\"false\">x_0\\in I.<\/span> Cela peut \u00eatre r\u00e9sum\u00e9 math\u00e9matiquement par l&#8217;expression suivante.<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\forall x_0 \\in I\\subseteq Dom(f) \\right)\\left(\\forall \\epsilon \\gt 0 \\right) \\left(\\exists \\delta \\gt 0 \\right) \\left(0\\lt |x-x_0|\\lt \\delta \\rightarrow |f(x) - f(x_0)|\\lt \\epsilon\\right)<\/span>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Propri\u00e9t\u00e9s des Fonctions Continues<\/h3>\n<h4>Alg\u00e8bre des Fonctions Continues<\/h4>\n<p ><a href=\"https:\/\/www.youtube.com\/watch?v=4kehfJcyPRg&amp;t=575s\" target=\"_blank\" rel=\"noopener\"><strong>Si <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> sont des fonctions continues sur <span class=\"katex-eq\" data-katex-display=\"false\">I\\subseteq Dom(f)\\cap Dom(g)<\/span><\/strong><\/a>, alors nous avons :<\/p>\n<ul >\n<li><span class=\"katex-eq\" data-katex-display=\"false\">f\\pm g<\/span> est continue sur <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">f\\cdot g<\/span> est continue sur <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span><\/li>\n<li>Si <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha\\in\\mathbb{R}<\/span>, alors <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha f<\/span> est continue sur <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">f\/g<\/span> est continue sur <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span>, tant que <span class=\"katex-eq\" data-katex-display=\"false\">g\\neq 0<\/span><\/li>\n<li>Si <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha,\\beta\\in\\mathbb{R}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\beta\\neq 0<\/span>, alors <span class=\"katex-eq\" data-katex-display=\"false\">f^{\\alpha\/\\beta}<\/span> est continue sur <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span>, \u00e0 condition que <span class=\"katex-eq\" data-katex-display=\"false\">f^{\\alpha\/\\beta}<\/span> soit bien d\u00e9finie sur <span class=\"katex-eq\" data-katex-display=\"false\">I.<\/span><\/li>\n<\/ul>\n<p >Toutes ces propri\u00e9t\u00e9s, qui semblent difficiles \u00e0 d\u00e9montrer \u00e0 partir de la d\u00e9finition formelle de la continuit\u00e9, sont en r\u00e9alit\u00e9 tr\u00e8s faciles car elles sont pratiquement analogues \u00e0 l&#8217;alg\u00e8bre des limites que nous avons <a href=\"https:\/\/toposuranos.com\/la-definicion-de-limite-demostraciones-y-teoremas\/\" rel=\"noopener\" target=\"_blank\">d\u00e9j\u00e0 d\u00e9montr\u00e9e<\/a>, ce qui nous dispense de les d\u00e9montrer \u00e0 nouveau.<\/p>\n<h5>Composition des Fonctions Continues<\/h5>\n<p >Si <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> est continue sur <span class=\"katex-eq\" data-katex-display=\"false\">I\\subseteq Dom(f)<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> est continue sur <span class=\"katex-eq\" data-katex-display=\"false\">f(I)<\/span>, alors la composition <span class=\"katex-eq\" data-katex-display=\"false\">(g\\circ f)(x) = g(f(x))<\/span> est continue sur <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span>. Cela, comme on peut s&#8217;y attendre, est une cons\u00e9quence directe des lois de composition pour les limites des fonctions.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Exercices<\/h2>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/R91vswfQT_Q\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<ol >\n<li>Analysez la continuit\u00e9 des fonctions suivantes<br \/>\n<table>\n<tbody>\n<tr>\n<td width=\"30\">a.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\dfrac{1}{x-2} - 3x<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=33s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>b.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\dfrac{x+1}{x^2-4x+3}<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=97s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>c.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=|x-1| + \\sin(x)<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=215s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>d.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\dfrac{\\cos(x)}{x}<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=450s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>e.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=csc(2x)<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=564s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>f.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\dfrac{x\\tan(x)}{x^2 + 1}<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=666s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>g.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\sqrt{2x + 3}<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=784s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>h.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\dfrac{x+2}{\\cos(x)}<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=917s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>i.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\dfrac{\\sqrt{x^4 + 1}}{1+\\sin^2(x)}<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=952s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Calculez les limites suivantes. Les fonctions sont-elles continues aux points auxquels elles tendent ?<br \/>\n<table>\n<tbody>\n<tr>\n<td width=\"30\">a.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to\\pi} \\sin(x -\\sin(x))<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=1033s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>b.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0} \\sin\\left(\\dfrac{\\pi}{2}\\cos(\\tan(x)) \\right)<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=1092s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>c.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 1} \\sec\\left(x\\sec^2(x) - \\tan^2(x) -1 \\right)<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=1162s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>d.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0} \\tan\\left(\\dfrac{\\pi}{4}\\cos\\left(\\sin\\left(x^{1\/3}\\right)\\right) \\right)<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=1274s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>e.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0} \\cos\\left(\\dfrac{\\pi}{\\sqrt{19 - 3\\sec{2x}}} \\right)<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=1424s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>f.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to \\pi\/6} \\sqrt{\\csc^2(x) + 5\\sqrt{3}\\tan(x)}<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=1505s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Limite et Continuit\u00e9 R\u00e9sum\u00e9 : Ce cours traite de la relation entre la limite et la continuit\u00e9 d&#8217;une fonction, en commen\u00e7ant par une explication intuitive et formelle du terme. La continuit\u00e9 en un point et sur un ensemble est explor\u00e9e, en d\u00e9taillant les conditions n\u00e9cessaires pour qu&#8217;une fonction soit continue. Les propri\u00e9t\u00e9s alg\u00e9briques des fonctions [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":29557,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":0,"footnotes":""},"categories":[866,569],"tags":[],"class_list":["post-29572","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>Limite et Continuit\u00e9 - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Qu&#039;est-ce que la continuit\u00e9 et quel est son rapport avec la limite ? 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