{"id":27994,"date":"2024-08-17T13:00:35","date_gmt":"2024-08-17T13:00:35","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27994"},"modified":"2024-08-16T02:00:03","modified_gmt":"2024-08-16T02:00:03","slug":"limites-laterales-definitions-developpement-et-exercices","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/fr\/limites-laterales-definitions-developpement-et-exercices\/","title":{"rendered":"Limites Lat\u00e9rales : D\u00e9finitions, D\u00e9veloppement et Exercices"},"content":{"rendered":"<p><center><\/p>\n<h1>Limites Lat\u00e9rales : D\u00e9finitions, D\u00e9veloppement et Exercices<\/h1>\n<p><em><strong>R\u00e9sum\u00e9 :<\/strong><br \/>\nCe post explique les limites lat\u00e9rales et bilat\u00e9rales en calcul, montrant comment elles sont d\u00e9finies et appliqu\u00e9es de mani\u00e8re intuitive et formelle. Des exemples graphiques et alg\u00e9briques sont pr\u00e9sent\u00e9s, les conditions d&#8217;utilisation de l&#8217;alg\u00e8bre des limites dans ces cas sont discut\u00e9es, et des exercices r\u00e9solus sont inclus pour renforcer la compr\u00e9hension. L&#8217;objectif est de fournir une vision claire et concise de ces concepts fondamentaux dans l&#8217;\u00e9tude du calcul.<\/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<ul style=\"text-align:left;\">\n<li><strong>Comprendre<\/strong> la diff\u00e9rence entre les limites lat\u00e9rales et bilat\u00e9rales.<\/li>\n<li><strong>D\u00e9finir<\/strong> formellement les limites lat\u00e9rales \u00e0 droite et \u00e0 gauche.<\/li>\n<li><strong>Appliquer<\/strong> la d\u00e9finition des limites lat\u00e9rales \u00e0 des probl\u00e8mes de calcul.<\/li>\n<li><strong>Interpr\u00e9ter<\/strong> graphiquement les limites lat\u00e9rales et bilat\u00e9rales.<\/li>\n<li><strong>Reconna\u00eetre<\/strong> les conditions n\u00e9cessaires pour qu&#8217;une limite bilat\u00e9rale existe.<\/li>\n<li><strong>Utiliser<\/strong> l&#8217;alg\u00e8bre des limites dans des contextes de limites lat\u00e9rales.<\/li>\n<li><strong>R\u00e9soudre<\/strong> des exercices de calcul li\u00e9s aux limites lat\u00e9rales.<\/li>\n<\/ul>\n<p><strong>TABLE DES MATI\u00c8RES :<\/strong><br \/>\n<a href=\"#1\">Introduction<\/a><br \/>\n<a href=\"#2\">Id\u00e9e Intuitive des Limites Lat\u00e9rales et Bilat\u00e9rales<\/a><br \/>\n<a href=\"#3\">D\u00e9finition Formelle des Limites Lat\u00e9rales<\/a><br \/>\n<a href=\"#4\">Conditions pour l&#8217;Alg\u00e8bre des Limites<\/a><br \/>\n<a href=\"#5\">Exercices Propos\u00e9s et R\u00e9solus<\/a><\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/8lkvxEHsF6c\" title=\"Lecteur vid\u00e9o YouTube\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\n<\/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=8lkvxEHsF6c&amp;t=138s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Les limites lat\u00e9rales apparaissent lorsque nous rencontrons des limites qui ne peuvent exister que de gauche ou de droite,<\/span><\/strong><\/a> mais pas des deux c\u00f4t\u00e9s. Les limites que nous avons \u00e9tudi\u00e9es jusqu&#8217;\u00e0 pr\u00e9sent sont pr\u00e9cis\u00e9ment de ce dernier type : pour que la limite de la fonction <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> lorsque <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\to x_0<\/span><\/span> existe, il est n\u00e9cessaire que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> soit bien d\u00e9finie de chaque c\u00f4t\u00e9 de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>; si ce n&#8217;est pas le cas, alors la d\u00e9finition de la limite ne fonctionnera pas. Comme ces types de situations sont fr\u00e9quents, il est n\u00e9cessaire de trouver un moyen de les traiter. Cela est r\u00e9solu par une d\u00e9finition formelle.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Id\u00e9e Intuitive des Limites Lat\u00e9rales et Bilat\u00e9rales<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=279s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Pour qu&#8217;une limite de fonction existe<\/span><\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span>, lorsque <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\to x_0<\/span><\/span>, il est n\u00e9cessaire que la fonction soit bien d\u00e9finie de chaque c\u00f4t\u00e9 de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>. Si cela se produit, nous parlons d&#8217;une limite bilat\u00e9rale. Et si cette limite donne le r\u00e9sultat <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">L<\/span><\/span>, alors il n&#8217;y aura aucun probl\u00e8me \u00e0 \u00e9crire<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}f(x) = L<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Maintenant, imaginons que nous red\u00e9finissions cette fonction de mani\u00e8re \u00e0 ce que son domaine inclue uniquement des valeurs sup\u00e9rieures \u00e0 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>. Si nous faisons cela, nous remarquerons que la limite n&#8217;existe plus (car il existera des valeurs de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> pour lesquelles cela n&#8217;aura pas de sens); cependant, graphiquement, nous pourrions encore dire que, lorsque <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\to x_0<\/span><\/span>, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> tend toujours vers <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">L<\/span><\/span>. L&#8217;id\u00e9e intuitive que nous g\u00e9n\u00e9rons ici est celle d&#8217;une limite \u00e0 droite, que nous repr\u00e9senterions par l&#8217;\u00e9criture<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0^+}f(x) = L<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">et de mani\u00e8re compl\u00e8tement analogue, nous aurons la limite \u00e0 gauche<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0^-}f(x) = L<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Enfin, la limite bilat\u00e9rale existera tant que les limites lat\u00e9rales existent et sont \u00e9gales<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0^-}f(x) = \\lim_{x\\to x_0}f(x) = \\lim_{x\\to x_0^+}f(x)<\/span><\/span><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>D\u00e9finition Formelle des Limites Lat\u00e9rales<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=438s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Pour d\u00e9finir formellement les limites lat\u00e9rales, il suffit<\/span> <\/strong><\/a>d&#8217;appliquer une petite modification \u00e0 la d\u00e9finition originale de la limite.<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} f(x) = L := \\left(\\forall \\epsilon\\gt 0 \\right)<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\exist \\delta \\gt 0 \\right)<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\right.<\/span><\/span> <span style=\"background-color: #88ff88;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\lt|x-x_0|\\lt\\delta <\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left.\\rightarrow |f(x)-L|\\lt\\epsilon \\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Pour les limites \u00e0 droite, la d\u00e9finition est la suivante :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\forall \\epsilon\\gt 0 \\right)<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\exist \\delta \\gt 0 \\right)<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\right.<\/span><\/span> <span style=\"background-color: #88ff88;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0 \\lt x \\lt x_0 + \\delta <\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left.\\rightarrow |f(x)-L|\\lt\\epsilon \\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Pour les limites \u00e0 gauche, elle sera :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\forall \\epsilon\\gt 0 \\right)<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\exist \\delta \\gt 0 \\right)<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\right.<\/span><\/span> <span style=\"background-color: #88ff88;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0 - \\delta \\lt x \\lt x_0<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left.\\rightarrow |f(x)-L|\\lt\\epsilon \\right)<\/span><\/span><\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Conditions pour l&#8217;Alg\u00e8bre des Limites<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=801s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">L&#8217;int\u00e9r\u00eat de ces d\u00e9finitions est qu&#8217;elles sont toutes deux contenues<\/span><\/strong><\/a> dans la d\u00e9finition habituelle de la limite, et cela est important car cela nous dispense de devoir prouver \u00e0 nouveau toutes les propri\u00e9t\u00e9s que nous avons d\u00e9j\u00e0 prouv\u00e9es pour les limites bilat\u00e9rales. Toute l&#8217;alg\u00e8bre des limites fonctionnera comme nous l&#8217;avons vu dans les cours pr\u00e9c\u00e9dents, tant que les limites impliqu\u00e9es sont de m\u00eame nature (toutes deux \u00e0 gauche, ou toutes deux \u00e0 droite, jamais m\u00e9lang\u00e9es), dirig\u00e9es vers le m\u00eame point et existent \u00e0 ce point.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Exercices Propos\u00e9s et R\u00e9solus<\/h2>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to {\\frac{1}{2}}^- } \\sqrt{\\dfrac{x+2}{x+1}} <\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=994s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 1^+} \\sqrt{\\dfrac{x-1}{x+2}} <\/span><\/span><a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=1058s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"> <strong>[SOLUTION]<\/strong><\/span><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 2^+} \\left(\\dfrac{x}{x+1} \\right) \\left(\\dfrac{2x+5}{x^2+x} \\right)<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=1058s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[SOLUTION]<\/strong><\/span><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 1^-} \\left(\\dfrac{1}{x+1} \\right) \\left(\\dfrac{x+6}{x} \\right) \\left(\\dfrac{3-x}{x} \\right) <\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=1102s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[SOLUTION]<\/strong><\/span><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{h\\to 0^+ } \\dfrac{\\sqrt{h^2 + 4h +5} - \\sqrt{5}}{h} <\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=1168s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[SOLUTION]<\/strong><\/span><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{h\\to 0^-} \\dfrac{\\sqrt{6} - \\sqrt{5h^2 + 11h + 6}}{h}<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=1276s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li>\n<table>\n<tbody>\n<tr style=\"text-align: justify;\">\n<td>a. <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to -2^+} (x+3)\\dfrac{|x+2|}{x+2}\n\n<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>b. <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to -2^-} (x+3)\\dfrac{|x+2|}{x+2}\n\n<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=1378s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li>\n<table>\n<tbody>\n<tr style=\"text-align: justify;\">\n<td>a. <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 1^+} \\dfrac{\\sqrt{2x}(x-1)}{|x-1|}\n\n<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>b. <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 1^-}\\dfrac{\\sqrt{2x}(x-1)}{|x-1|}\n\n<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=1655s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Limites Lat\u00e9rales : D\u00e9finitions, D\u00e9veloppement et Exercices R\u00e9sum\u00e9 : Ce post explique les limites lat\u00e9rales et bilat\u00e9rales en calcul, montrant comment elles sont d\u00e9finies et appliqu\u00e9es de mani\u00e8re intuitive et formelle. Des exemples graphiques et alg\u00e9briques sont pr\u00e9sent\u00e9s, les conditions d&#8217;utilisation de l&#8217;alg\u00e8bre des limites dans ces cas sont discut\u00e9es, et des exercices r\u00e9solus sont [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27972,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":71,"footnotes":""},"categories":[866,569],"tags":[],"class_list":["post-27994","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>Limites Lat\u00e9rales : D\u00e9finitions, D\u00e9veloppement et Exercices - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Explorez une explication d\u00e9taill\u00e9e des limites lat\u00e9rales et 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