Quiz — مراجعة على النهايات.pdf

Question 1 sur 10 20:00

[{"id":17979,"question":"Quelle est la limite de la fonction f(x) = (2x² - 3x + 1)\/(x² + 5) quand x tend vers +∞ ?","option_a":"A. +∞","option_b":"B. -∞","option_c":"C. 2","option_d":"D. 0","option_e":"","option_f":"","bonne_reponse":"c","explication":"En divisant numérateur et dénominateur par x², on obtient f(x) = (2 - 3\/x + 1\/x²)\/(1 + 5\/x²). Quand x→+∞, les termes en 1\/x tendent vers 0, donc la limite est 2\/1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A. +∞\", \"b\": \"B. -∞\", \"c\": \"C. 2\", \"d\": \"D. 0\"}}","_debug_options_count":4},{"id":17980,"question":"La fonction f(x) = (x² + 1)\/x admet une asymptote verticale en x=0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction n'est pas définie en x=0, mais lim(x→0) f(x) = ±∞ selon le signe de x. Elle admet donc une asymptote verticale en x=0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":17981,"question":"Quelle méthode permet de lever l'indétermination 0\/0 dans le calcul d'une limite ?","option_a":"A. Factoriser le numérateur et le dénominateur","option_b":"B. Appliquer le théorème des gendarmes","option_c":"C. Calculer la dérivée","option_d":"D. Utiliser les limites comparées","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour lever une indétermination 0\/0, on factorise généralement le numérateur et le dénominateur pour simplifier l'expression avant de calculer la limite.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. Factoriser le numérateur et le dénominateur\", \"b\": \"B. Appli","_debug_options_count":4},{"id":17982,"question":"Soit f(x) = (x³ - 2x + 1)\/(x² - 1). Quelle est la limite de f(x) quand x tend vers 1 ?","option_a":"A. 0","option_b":"B. +∞","option_c":"C. -∞","option_d":"D. 1\/2","option_e":"","option_f":"","bonne_reponse":"b","explication":"En x=1, on a une forme indéterminée 0\/0. En factorisant (x³ - 2x + 1) = (x-1)(x² + x - 1) et (x² - 1) = (x-1)(x+1), on simplifie f(x) = (x² + x - 1)\/(x+1). La limite quand x→1 est donc (1+1-1)\/(1+1) = 1\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. +∞\", \"c\": \"C. -∞\", \"d\": \"D. 1\/2\"}}","_debug_options_count":4},{"id":17983,"question":"Le théorème des gendarmes s'applique uniquement aux limites finies.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le théorème des gendarmes s'applique aussi bien aux limites finies qu'infinies. Il permet de déterminer la limite d'une fonction encadrée par deux autres fonctions dont les limites sont égales.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":17984,"question":"Quelle est la limite de la fonction f(x) = √(x² + 1) - x quand x tend vers +∞ ?","option_a":"A. 0","option_b":"B. +∞","option_c":"C. 1\/2","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"En multipliant par l'expression conjuguée, on obtient f(x) = 1\/(√(x² + 1) + x). Quand x→+∞, le dénominateur tend vers +∞, donc la limite est 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. +∞\", \"c\": \"C. 1\/2\", \"d\": \"D. -∞\"}}","_debug_options_count":4},{"id":17985,"question":"Soit f(x) = (sin(x))\/x. Quelle est la limite de f(x) quand x tend vers 0 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est une limite classique : lim(x→0) sin(x)\/x = 1 (théorème de la limite fondamentale).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. +∞\", \"d\": \"D. -∞\"}}","_debug_options_count":4},{"id":17986,"question":"La fonction f(x) = (x² + 3x - 4)\/(x - 1) admet une asymptote oblique.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En simplifiant f(x) = (x+4) pour x≠1, on voit que la droite y = x + 4 est une asymptote oblique à la courbe représentative de f.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":17987,"question":"Quelle est la limite de la fonction f(x) = (e^x - 1)\/x quand x tend vers 0 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. e","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est une limite classique : lim(x→0) (e^x - 1)\/x = 1 (dérivée de e^x en 0).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. e\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":17988,"question":"Soit f(x) = (x^3 - 1)\/(x - 1). Quelle est la limite de f(x) quand x tend vers 1 ?","option_a":"A. 0","option_b":"B. 3","option_c":"C. +∞","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"b","explication":"En factorisant (x^3 - 1) = (x-1)(x² + x + 1), on simplifie f(x) = x² + x + 1. La limite quand x→1 est donc 1 + 1 + 1 = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. 3\", \"c\": \"C. +∞\", \"d\": \"D. 1\"}}","_debug_options_count":4}]