Quiz — 654b3fb22adf8_énoncé_Série N°1-Limites et continuité.pdf

Question 1 sur 10 20:00

[{"id":30010,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 1)\/(x² - 4) quand x tend vers +∞ ?","option_a":"A. +∞","option_b":"B. -∞","option_c":"C. 3","option_d":"D. 0","option_e":"","option_f":"","bonne_reponse":"c","explication":"En divisant numérateur et dénominateur par x², on obtient f(x) = (3 + 2\/x - 1\/x²)\/(1 - 4\/x²). Quand x tend vers +∞, les termes en 1\/x et 1\/x² tendent vers 0, donc la limite est 3\/1 = 3.","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. 3\", \"d\": \"D. 0\"}}","_debug_options_count":4},{"id":30011,"question":"Une fonction discontinue en un point peut-elle avoir une limite en ce point ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, une fonction peut être discontinue en un point tout en ayant une limite en ce point (exemple : la fonction partie entière en un entier).","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":30012,"question":"Soit f une fonction continue sur [a, b]. Si f(a) = -2 et f(b) = 3, alors l'équation f(x) = 0 admet au moins une solution dans [a, b].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est le théorème des valeurs intermédiaires : comme f est continue et change de signe sur [a, b], elle s'annule nécessairement au moins une fois.","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":30013,"question":"Quelle est la limite de la fonction g(x) = (sin x)\/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, souvent démontrée par le théorème des gendarmes ou par développement limité.","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":30014,"question":"Si une fonction f est continue sur un intervalle fermé [a, b], alors elle est bornée sur cet intervalle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est le théorème des bornes atteintes : une fonction continue sur un segment fermé est bornée et atteint ses bornes.","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":30015,"question":"Quelle est la nature de l'asymptote de la fonction h(x) = (2x² + 3x - 1)\/(x + 2) quand x tend vers -∞ ?","option_a":"A. Horizontale","option_b":"B. Verticale","option_c":"C. Oblique","option_d":"D. Aucune asymptote","option_e":"","option_f":"","bonne_reponse":"c","explication":"En effectuant la division euclidienne, on obtient h(x) = 2x - 1 + 1\/(x + 2). La droite d'équation y = 2x - 1 est donc une asymptote oblique.","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. Horizontale\", \"b\": \"B. Verticale\", \"c\": \"C. Oblique\", \"d\": \"D.","_debug_options_count":4},{"id":30016,"question":"Soit f(x) = x² - 3x + 2. La fonction f est-elle continue en x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"f est une fonction polynôme, donc continue sur ℝ. Elle est donc continue en x = 1.","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":30017,"question":"Quelle est la limite de la fonction k(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, souvent démontrée par la définition de la 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":30018,"question":"Si lim (x→a) f(x) = L et lim (x→a) g(x) = M, alors lim (x→a) [f(x) + g(x)] = L + M.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une propriété fondamentale des limites : la limite d'une somme est la somme des limites, sous réserve que ces limites existent.","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":30019,"question":"Quelle est la limite de la fonction m(x) = (ln x)\/(x - 1) quand x tend vers 1 ?","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 forme indéterminée 0\/0. En utilisant le théorème de l'Hôpital (dérivées), on obtient lim (x→1) (1\/x)\/1 = 1.","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}]