{"id":1531,"date":"2026-09-08T08:08:38","date_gmt":"2026-09-08T08:08:38","guid":{"rendered":"https:\/\/maroczain.com\/scolaire.maroczain.com\/?page_id=1531"},"modified":"2026-09-08T08:08:38","modified_gmt":"2026-09-08T08:08:38","slug":"approcher-simplifier-comparer","status":"publish","type":"page","link":"https:\/\/maroczain.com\/scolaire.maroczain.com\/approcher-simplifier-comparer\/","title":{"rendered":"Approcher. simplifier. Comparer"},"content":{"rendered":"\n&#8220;`html\n<div id=\"mza-dl-cpge\">\n\n<style>\n#mza-dl-cpge,#mza-dl-cpge *{box-sizing:border-box}\n#mza-dl-cpge{\n --navy:#06172c;--navy2:#0d416d;--gold:#c99a36;--gold2:#efd68c;\n --paper:#f5f6f8;--white:#fff;--text:#18283a;--muted:#687789;--line:#dfe5eb;\n font-family:Arial,Helvetica,sans-serif;background:var(--paper);color:var(--text);overflow:hidden\n}\n#mza-dl-cpge a{text-decoration:none;color:inherit}\n#mza-dl-cpge .wrap{max-width:1200px;margin:auto;padding:0 24px}\n#mza-dl-cpge section{padding:68px 0}\n#mza-dl-cpge .hero{\n color:#fff;padding:72px 0;\n background:\n radial-gradient(circle at 84% 15%,rgba(213,169,68,.28),transparent 28%),\n linear-gradient(135deg,#041224,#082b4c 65%,#10517e)\n}\n#mza-dl-cpge .back{display:inline-flex;color:#dbe6ee;font-size:12px;font-weight:900;margin-bottom:24px}\n#mza-dl-cpge .badge{\n display:inline-block;padding:8px 13px;border-radius:999px;\n border:1px solid rgba(239,214,140,.4);color:var(--gold2);\n font-size:11px;font-weight:900;letter-spacing:1.3px\n}\n#mza-dl-cpge h1{font:500 clamp(42px,7vw,72px)\/1 Georgia,serif;margin:17px 0;color:#fff}\n#mza-dl-cpge .lead{max-width:880px;color:#d5e1ea;font-size:18px;line-height:1.75}\n#mza-dl-cpge .actions{display:flex;gap:9px;flex-wrap:wrap;margin-top:26px}\n#mza-dl-cpge .btn{\n min-height:47px;padding:0 18px;border:none;border-radius:10px;\n display:inline-flex;align-items:center;justify-content:center;cursor:pointer;\n font-size:12px;font-weight:900\n}\n#mza-dl-cpge .gold{background:linear-gradient(135deg,#bd892b,#efd483);color:#142238}\n#mza-dl-cpge .glass{background:rgba(255,255,255,.08);border:1px solid rgba(255,255,255,.18);color:#fff}\n#mza-dl-cpge .head{max-width:870px;margin-bottom:32px}\n#mza-dl-cpge .kicker{color:#987023;font-size:11px;font-weight:900;letter-spacing:1.3px}\n#mza-dl-cpge h2{color:var(--navy);font:500 clamp(31px,4vw,48px)\/1.1 Georgia,serif;margin:8px 0 11px}\n#mza-dl-cpge .head p{color:var(--muted);line-height:1.7}\n#mza-dl-cpge .grid{display:grid;grid-template-columns:repeat(2,1fr);gap:16px}\n#mza-dl-cpge .card{\n background:#fff;border:1px solid var(--line);border-radius:19px;padding:23px\n}\n#mza-dl-cpge .card h3{color:var(--navy);margin:0 0 10px;font-size:20px}\n#mza-dl-cpge .card p,#mza-dl-cpge .card li{color:var(--muted);line-height:1.7}\n#mza-dl-cpge .formula{\n margin:14px 0;padding:15px;border-left:4px solid var(--gold);\n background:#f8f6ef;border-radius:0 12px 12px 0;font-family:Georgia,serif;\n color:#213349;line-height:1.7\n}\n#mza-dl-cpge .tip{\n margin-top:14px;padding:13px;border-radius:11px;background:#eef3f7;\n color:#40546a;font-size:13px;line-height:1.6\n}\n#mza-dl-cpge .dark{background:var(--navy);color:#fff}\n#mza-dl-cpge .dark h2{color:#fff}\n#mza-dl-cpge .dark .head p{color:#c6d4df}\n#mza-dl-cpge .method{display:grid;grid-template-columns:repeat(6,1fr);gap:10px}\n#mza-dl-cpge .method div{\n padding:20px 10px;text-align:center;border-radius:14px;\n background:rgba(255,255,255,.07);border:1px solid rgba(255,255,255,.1)\n}\n#mza-dl-cpge .method b{display:block;color:var(--gold2);font-size:23px;margin-bottom:7px}\n#mza-dl-cpge .method span{font-size:10px;font-weight:900}\n#mza-dl-cpge .exercise{\n background:#fff;border:1px solid var(--line);border-radius:18px;padding:22px;margin-bottom:14px\n}\n#mza-dl-cpge .exercise h3{color:var(--navy);margin:0 0 10px}\n#mza-dl-cpge .level{\n display:inline-block;padding:6px 9px;margin-bottom:10px;border-radius:999px;\n background:#f1eee5;color:#7b5b1d;font-size:10px;font-weight:900\n}\n#mza-dl-cpge .exercise p{color:var(--muted);line-height:1.7}\n#mza-dl-cpge .hint{\n display:none;margin-top:12px;padding:15px;border-radius:12px;\n background:#eef3f7;color:#425469;line-height:1.65\n}\n#mza-dl-cpge .qcm{\n background:#fff;border:1px solid var(--line);border-radius:23px;padding:28px\n}\n#mza-dl-cpge .question{padding:20px;margin:15px 0;background:#f7f8fa;border-radius:14px}\n#mza-dl-cpge .question strong{display:block;color:var(--navy);margin-bottom:12px}\n#mza-dl-cpge label{display:block;padding:8px 0;color:#526174;cursor:pointer}\n#mza-dl-cpge #dlResult{\n display:none;margin-top:18px;padding:20px;border-radius:14px;background:var(--navy);color:#fff\n}\n#mza-dl-cpge .cta{\n text-align:center;padding:50px 24px;border-radius:25px;\n background:linear-gradient(135deg,#06172c,#0d416d);color:#fff\n}\n#mza-dl-cpge .cta h2{color:#fff}\n#mza-dl-cpge .cta p{max-width:720px;margin:0 auto 22px;color:#cfdae4;line-height:1.7}\n@media(max-width:900px){\n #mza-dl-cpge .grid,#mza-dl-cpge .method{grid-template-columns:1fr}\n}\n<\/style>\n\n<header class=\"hero\">\n<div class=\"wrap\">\n<a class=\"back\" href=\"https:\/\/maroczain.com\/scolaire.maroczain.com\/mathematiques-cpge\/\">\u2190 Math\u00e9matiques CPGE<\/a>\n<br>\n<span class=\"badge\">COURS 06 \u2022 D\u00c9VELOPPEMENTS LIMIT\u00c9S<\/span>\n\n<h1>Approcher.<br>Simplifier. Comparer.<\/h1>\n\n<p class=\"lead\">\nLes d\u00e9veloppements limit\u00e9s permettent de remplacer localement une fonction compliqu\u00e9e\npar un polyn\u00f4me simple. Ils sont essentiels pour les limites, les \u00e9quivalents,\nl&#8217;\u00e9tude locale des courbes et de nombreux calculs de concours.\n<\/p>\n\n<div class=\"actions\">\n<a href=\"#cours\" class=\"btn gold\">\ud83d\udcda COMMENCER<\/a>\n<a href=\"#methode\" class=\"btn glass\">\ud83e\udde0 M\u00c9THODE<\/a>\n<a href=\"#exercices\" class=\"btn glass\">\ud83e\udde9 EXERCICES<\/a>\n<a href=\"#qcm\" class=\"btn glass\">\u2753 QCM<\/a>\n<\/div>\n<\/div>\n<\/header>\n\n<section id=\"cours\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">1. PRINCIPE<\/span>\n<h2>Remplacer une fonction par un polyn\u00f4me local.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>D\u00e9veloppement limit\u00e9 en 0<\/h3>\n<p>\nOn cherche une \u00e9criture de la forme :\n<\/p>\n\n<div class=\"formula\">\nf(x)=a\u2080+a\u2081x+a\u2082x\u00b2+&#8230;+a\u2099x\u207f+o(x\u207f)\n<\/div>\n\n<p>\nlorsque x tend vers 0.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Petit o<\/h3>\n\n<div class=\"formula\">\nr(x)=o(x\u207f)\n<\/div>\n\n<p>\nsignifie que :\n<\/p>\n\n<div class=\"formula\">\nr(x)\/x\u207f \u2192 0\n<\/div>\n\n<p>lorsque x \u2192 0.<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Ordre<\/h3>\n<p>\nUn DL \u00e0 l&#8217;ordre n conserve tous les termes jusqu&#8217;\u00e0 x\u207f.\n<\/p>\n\n<div class=\"tip\">\nL&#8217;ordre \u00e0 choisir d\u00e9pend de la pr\u00e9cision n\u00e9cessaire dans le calcul.\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>DL en un point a<\/h3>\n\n<p>\nOn pose souvent :\n<\/p>\n\n<div class=\"formula\">\nh=x-a\n<\/div>\n\n<p>\npuis on d\u00e9veloppe en h au voisinage de 0.\n<\/p>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">2. FORMULE DE TAYLOR<\/span>\n<h2>Construire les coefficients du DL.<\/h2>\n<\/div>\n\n<div class=\"card\">\n\n<p>\nSi f poss\u00e8de suffisamment de d\u00e9riv\u00e9es au voisinage de a, alors :\n<\/p>\n\n<div class=\"formula\">\nf(x)=f(a)\n+f'(a)(x-a)\n+f&#8221;(a)\/2! \u00b7 (x-a)\u00b2\n+&#8230;+\nf\u207d\u207f\u207e(a)\/n! \u00b7 (x-a)\u207f\n+o((x-a)\u207f)\n<\/div>\n\n<div class=\"tip\">\nEn pratique, on apprend surtout les DL usuels et on les combine intelligemment.\n<\/div>\n\n<\/div>\n\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">3. DL USUELS EN 0<\/span>\n<h2>Les formules \u00e0 conna\u00eetre.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Exponentielle<\/h3>\n<div class=\"formula\">\ne\u02e3 = 1 + x + x\u00b2\/2 + x\u00b3\/6 + o(x\u00b3)\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Logarithme<\/h3>\n<div class=\"formula\">\nln(1+x)=x-x\u00b2\/2+x\u00b3\/3+o(x\u00b3)\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Sinus<\/h3>\n<div class=\"formula\">\nsin x = x &#8211; x\u00b3\/6 + o(x\u00b3)\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Cosinus<\/h3>\n<div class=\"formula\">\ncos x = 1 &#8211; x\u00b2\/2 + x\u2074\/24 + o(x\u2074)\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Inverse<\/h3>\n<div class=\"formula\">\n1\/(1+x)=1-x+x\u00b2-x\u00b3+o(x\u00b3)\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Puissance r\u00e9elle<\/h3>\n<div class=\"formula\">\n(1+x)<sup>\u03b1<\/sup>\n=1+\u03b1x+\u03b1(\u03b1-1)x\u00b2\/2+o(x\u00b2)\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section id=\"methode\" class=\"dark\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\" style=\"color:#efd68c\">MZA ASYMPTOTIC ENGINE<\/span>\n<h2>La m\u00e9thode CPGE.<\/h2>\n<\/div>\n\n<div class=\"method\">\n<div><b>01<\/b><span>IDENTIFIER LE POINT<\/span><\/div>\n<div><b>02<\/b><span>CHOISIR L&#8217;ORDRE<\/span><\/div>\n<div><b>03<\/b><span>UTILISER UN DL USUEL<\/span><\/div>\n<div><b>04<\/b><span>COMPOSER<\/span><\/div>\n<div><b>05<\/b><span>SIMPLIFIER<\/span><\/div>\n<div><b>06<\/b><span>CONCLURE<\/span><\/div>\n<\/div>\n\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">4. OP\u00c9RATIONS SUR LES DL<\/span>\n<h2>Addition, produit, quotient, composition.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Somme<\/h3>\n\n<p>\nOn additionne terme \u00e0 terme les d\u00e9veloppements jusqu&#8217;\u00e0 l&#8217;ordre souhait\u00e9.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Produit<\/h3>\n\n<p>\nOn d\u00e9veloppe puis on conserve uniquement les puissances utiles.\n<\/p>\n\n<div class=\"tip\">\nNe gardez pas des termes d&#8217;ordre sup\u00e9rieur \u00e0 celui demand\u00e9.\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Quotient<\/h3>\n\n<p>\nOn transforme souvent le d\u00e9nominateur sous la forme :\n<\/p>\n\n<div class=\"formula\">\n1+u(x)\n<\/div>\n\n<p>\npuis on utilise le DL de 1\/(1+u).\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Composition<\/h3>\n\n<p>\nSi u(x) \u2192 0, on peut remplacer x par u(x) dans un DL usuel.\n<\/p>\n\n<div class=\"formula\">\ne<sup>u(x)<\/sup>\n=1+u(x)+u(x)\u00b2\/2+&#8230;\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">5. DL &#038; LIMITES<\/span>\n<h2>Lever les formes ind\u00e9termin\u00e9es.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Exemple-type<\/h3>\n\n<p>\u00c9tudier :<\/p>\n\n<div class=\"formula\">\n(e\u02e3-1-x)\/x\u00b2\n<\/div>\n\n<p>au voisinage de 0.<\/p>\n\n<div class=\"tip\">\nLe DL d&#8217;e\u02e3 permet d&#8217;identifier imm\u00e9diatement le premier terme non nul du num\u00e9rateur.\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Premier terme non nul<\/h3>\n\n<p>\nPour une limite de quotient, le premier terme non nul est souvent celui qui d\u00e9termine le comportement principal.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>\u00c9quivalent issu du DL<\/h3>\n\n<p>\nSi :\n<\/p>\n\n<div class=\"formula\">\nf(x)=ax\u1d56+o(x\u1d56), a \u2260 0\n<\/div>\n\n<p>\nalors :\n<\/p>\n\n<div class=\"formula\">\nf(x) ~ ax\u1d56\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Ordre suffisant<\/h3>\n\n<p>\nIl faut parfois d\u00e9velopper plus loin lorsque les premiers termes se compensent.\n<\/p>\n\n<div class=\"tip\">\nUne compensation compl\u00e8te des premiers termes est un signal : augmentez l&#8217;ordre du DL.\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">6. INTERPR\u00c9TATION G\u00c9OM\u00c9TRIQUE<\/span>\n<h2>Tangente et position locale.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>DL d&#8217;ordre 1<\/h3>\n\n<div class=\"formula\">\nf(a+h)=f(a)+f'(a)h+o(h)\n<\/div>\n\n<p>\nLe polyn\u00f4me de degr\u00e9 1 correspond \u00e0 la tangente.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Position par rapport \u00e0 la tangente<\/h3>\n\n<p>\nLe signe du premier terme non nul apr\u00e8s le terme affine indique localement la position de la courbe.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Exemple structurel<\/h3>\n\n<div class=\"formula\">\nf(a+h)=f(a)+f'(a)h+Ah\u00b2+o(h\u00b2)\n<\/div>\n\n<p>\nLe signe de A informe sur la position locale de la courbe par rapport \u00e0 sa tangente.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Courbure<\/h3>\n\n<p>\nLorsque f&#8221;(a) \u2260 0, le terme quadratique traduit la courbure locale.\n<\/p>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section id=\"exercices\" style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">7. EXERCICES PROGRESSIFS<\/span>\n<h2>Automatiser les calculs asymptotiques.<\/h2>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU 1<\/span>\n<h3>Exercice 1 \u2014 DL de base<\/h3>\n\n<p>Donner le DL \u00e0 l&#8217;ordre 3 en 0 de :<\/p>\n\n<div class=\"formula\">\ne\u02e3\n<\/div>\n\n<button class=\"btn gold\" type=\"button\" onclick=\"mzaDLHint('dlHint1')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"dlHint1\" class=\"hint\">\nUtilisez directement le d\u00e9veloppement usuel de l&#8217;exponentielle.\n<\/div>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU 2<\/span>\n<h3>Exercice 2 \u2014 Composition<\/h3>\n\n<p>Donner un DL \u00e0 l&#8217;ordre 4 en 0 de :<\/p>\n\n<div class=\"formula\">\ncos(x\u00b2)\n<\/div>\n\n<button class=\"btn gold\" type=\"button\" onclick=\"mzaDLHint('dlHint2')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"dlHint2\" class=\"hint\">\nRemplacez la variable du DL de cos par x\u00b2 et contr\u00f4lez l&#8217;ordre obtenu.\n<\/div>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU 3<\/span>\n<h3>Exercice 3 \u2014 Limite<\/h3>\n\n<p>\u00c9tudier :<\/p>\n\n<div class=\"formula\">\nlim [ln(1+x)-x]\/x\u00b2<br>\nx \u2192 0\n<\/div>\n\n<button class=\"btn gold\" type=\"button\" onclick=\"mzaDLHint('dlHint3')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"dlHint3\" class=\"hint\">\nD\u00e9veloppez ln(1+x) suffisamment loin pour faire appara\u00eetre le premier terme restant apr\u00e8s la soustraction de x.\n<\/div>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU PR\u00c9PA<\/span>\n<h3>Exercice 4 \u2014 Compensation<\/h3>\n\n<p>\u00c9tudier :<\/p>\n\n<div class=\"formula\">\nlim [e\u02e3 + e<sup>-x<\/sup> &#8211; 2 &#8211; x\u00b2] \/ x\u2074<br>\nx \u2192 0\n<\/div>\n\n<button class=\"btn gold\" type=\"button\" onclick=\"mzaDLHint('dlHint4')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"dlHint4\" class=\"hint\">\nLes termes impairs vont se compenser. Il faut donc d\u00e9velopper jusqu&#8217;\u00e0 l&#8217;ordre 4.\n<\/div>\n<\/div>\n\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">8. PROBLEM LAB<\/span>\n<h2>Mission : position par rapport \u00e0 la tangente.<\/h2>\n<\/div>\n\n<div class=\"card\">\n\n<h3>Mission CPGE<\/h3>\n\n<p>\nOn consid\u00e8re :\n<\/p>\n\n<div class=\"formula\">\nf(x)=ln(1+x)\n<\/div>\n\n<p>\nau voisinage de 0.\n<\/p>\n\n<p>Votre mission :<\/p>\n\n<ol>\n<li>obtenir un DL \u00e0 l&#8217;ordre 3 ;<\/li>\n<li>identifier la tangente \u00e0 la courbe en 0 ;<\/li>\n<li>\u00e9tudier localement la position de la courbe par rapport \u00e0 cette tangente ;<\/li>\n<li>expliquer le r\u00f4le du premier terme non nul apr\u00e8s la partie affine.<\/li>\n<\/ol>\n\n<div class=\"actions\">\n<button class=\"btn gold\" type=\"button\" onclick=\"mzaDLHint('dlMission1')\">\ud83d\udca1 INDICE<\/button>\n<button class=\"btn\" style=\"background:#06172c;color:#fff\" type=\"button\" onclick=\"mzaDLHint('dlMission2')\">\ud83e\udde0 M\u00c9THODE<\/button>\n<\/div>\n\n<div id=\"dlMission1\" class=\"hint\">\nCommencez par s\u00e9parer la partie affine du DL et les termes d&#8217;ordre sup\u00e9rieur.\n<\/div>\n\n<div id=\"dlMission2\" class=\"hint\">\n\u00c9crivez f(x) &#8211; T(x), o\u00f9 T est l&#8217;\u00e9quation affine de la tangente.\nLe signe du premier terme non nul de cette diff\u00e9rence donne la position locale de la courbe.\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section id=\"qcm\" style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">9. QCM DE VALIDATION<\/span>\n<h2>Diagnostic d\u00e9veloppements limit\u00e9s.<\/h2>\n<p>Le score est affich\u00e9, mais la correction reste masqu\u00e9e.<\/p>\n<\/div>\n\n<div class=\"qcm\">\n<form id=\"dlQuiz\">\n\n<div class=\"question\">\n<strong>1. Un DL \u00e0 l&#8217;ordre n conserve :<\/strong>\n<label><input type=\"radio\" name=\"d1\" value=\"1\"> les termes jusqu&#8217;\u00e0 la puissance n<\/label>\n<label><input type=\"radio\" name=\"d1\" value=\"0\"> uniquement le terme constant<\/label>\n<label><input type=\"radio\" name=\"d1\" value=\"0\"> uniquement les puissances paires<\/label>\n<label><input type=\"radio\" name=\"d1\" value=\"0\"> tous les termes sans limite<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>2. Au voisinage de 0, e\u02e3 &#8211; 1 est \u00e9quivalent \u00e0 :<\/strong>\n<label><input type=\"radio\" name=\"d2\" value=\"1\"> x<\/label>\n<label><input type=\"radio\" name=\"d2\" value=\"0\"> x\u00b2<\/label>\n<label><input type=\"radio\" name=\"d2\" value=\"0\"> 1\/x<\/label>\n<label><input type=\"radio\" name=\"d2\" value=\"0\"> 1<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>3. Une compensation des premiers termes signifie souvent qu&#8217;il faut :<\/strong>\n<label><input type=\"radio\" name=\"d3\" value=\"1\"> d\u00e9velopper \u00e0 un ordre sup\u00e9rieur<\/label>\n<label><input type=\"radio\" name=\"d3\" value=\"0\"> arr\u00eater imm\u00e9diatement<\/label>\n<label><input type=\"radio\" name=\"d3\" value=\"0\"> remplacer x par 1<\/label>\n<label><input type=\"radio\" name=\"d3\" value=\"0\"> conclure que la limite vaut 0<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>4. Si f(x)=ax\u00b2+o(x\u00b2), a\u22600, alors :<\/strong>\n<label><input type=\"radio\" name=\"d4\" value=\"1\"> f(x) ~ ax\u00b2<\/label>\n<label><input type=\"radio\" name=\"d4\" value=\"0\"> f(x) ~ x<\/label>\n<label><input type=\"radio\" name=\"d4\" value=\"0\"> f(x) ~ 1<\/label>\n<label><input type=\"radio\" name=\"d4\" value=\"0\"> f(x) est constante<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>5. Le DL d&#8217;ordre 1 en a fournit notamment :<\/strong>\n<label><input type=\"radio\" name=\"d5\" value=\"1\"> l&#8217;approximation affine et la tangente<\/label>\n<label><input type=\"radio\" name=\"d5\" value=\"0\"> toutes les racines de la fonction<\/label>\n<label><input type=\"radio\" name=\"d5\" value=\"0\"> son int\u00e9grale exacte<\/label>\n<label><input type=\"radio\" name=\"d5\" value=\"0\"> sa p\u00e9riode<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>6. Pour composer un DL usuel, il faut en particulier v\u00e9rifier que :<\/strong>\n<label><input type=\"radio\" name=\"d6\" value=\"1\"> la quantit\u00e9 substitu\u00e9e tend vers le point de d\u00e9veloppement<\/label>\n<label><input type=\"radio\" name=\"d6\" value=\"0\"> la fonction est forc\u00e9ment polynomiale<\/label>\n<label><input type=\"radio\" name=\"d6\" value=\"0\"> x est entier<\/label>\n<label><input type=\"radio\" name=\"d6\" value=\"0\"> la d\u00e9riv\u00e9e est nulle<\/label>\n<\/div>\n\n<button type=\"button\" class=\"btn gold\" onclick=\"mzaDLScore()\">VALIDER MON QCM<\/button>\n<\/form>\n\n<div id=\"dlResult\"><\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"cta\">\n<div style=\"font-size:43px\">\ud83d\udd2c<\/div>\n\n<h2>Voir l&#8217;essentiel derri\u00e8re une fonction compliqu\u00e9e.<\/h2>\n\n<p>\nUn d\u00e9veloppement limit\u00e9 permet d&#8217;identifier le comportement dominant,\nde lever des formes ind\u00e9termin\u00e9es et de comprendre localement une courbe\navec une grande pr\u00e9cision.\n<\/p>\n\n<div class=\"actions\" style=\"justify-content:center\">\n<a href=\"#cours\" class=\"btn gold\">\ud83d\udcda REVOIR LE COURS<\/a>\n<a href=\"#exercices\" class=\"btn glass\">\ud83e\udde9 EXERCICES<\/a>\n<a href=\"#qcm\" class=\"btn glass\">\u2753 QCM<\/a>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<script>\nfunction mzaDLHint(id){\n const el=document.getElementById(id);\n if(!el)return;\n el.style.display = el.style.display===\"block\" ? \"none\" : \"block\";\n}\n\nfunction mzaDLScore(){\n const form=document.getElementById(\"dlQuiz\");\n const result=document.getElementById(\"dlResult\");\n let score=0;\n let complete=true;\n\n [\"d1\",\"d2\",\"d3\",\"d4\",\"d5\",\"d6\"].forEach(function(name){\n   const answer=form.querySelector('input[name=\"'+name+'\"]:checked');\n   if(!answer){complete=false}\n   else{score+=Number(answer.value)}\n });\n\n result.style.display=\"block\";\n\n if(!complete){\n   result.innerHTML=\"<strong>R\u00e9pondez aux 6 questions avant de valider.<\/strong>\";\n   return;\n }\n\n const pct=Math.round(score\/6*100);\n let level=\"Calcul asymptotique \u00e0 renforcer\";\n\n if(pct>=50) level=\"Bases acquises\";\n if(pct>=67) level=\"Bon niveau\";\n if(pct>=84) level=\"Tr\u00e8s bonne ma\u00eetrise\";\n if(pct===100) level=\"Excellent niveau CPGE\";\n\n result.innerHTML=\n \"<strong style='font-size:30px;color:#efd68c'>\"+pct+\"%<\/strong>\"+\n \"<p><b>\"+level+\"<\/b><\/p>\"+\n \"<p>Score : \"+score+\" \/ 6<\/p>\"+\n \"<p>Les r\u00e9ponses correctes restent masqu\u00e9es.<\/p>\";\n}\n<\/script>\n\n<\/div>\n&#8220;`\n","protected":false},"excerpt":{"rendered":"<p>&#8220;`html \u2190 Math\u00e9matiques CPGE COURS 06 \u2022 D\u00c9VELOPPEMENTS LIMIT\u00c9S Approcher.Simplifier. Comparer. Les d\u00e9veloppements limit\u00e9s permettent de remplacer localement une fonction compliqu\u00e9e par un polyn\u00f4me simple. Ils sont essentiels pour les limites, les \u00e9quivalents, l&#8217;\u00e9tude locale des courbes et de nombreux calculs de concours. \ud83d\udcda COMMENCER \ud83e\udde0 M\u00c9THODE \ud83e\udde9 EXERCICES \u2753 QCM 1. PRINCIPE Remplacer une [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-1531","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages\/1531","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/comments?post=1531"}],"version-history":[{"count":1,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages\/1531\/revisions"}],"predecessor-version":[{"id":1533,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages\/1531\/revisions\/1533"}],"wp:attachment":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/media?parent=1531"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}