Quiz — Calculs de limites et d’équivalents, V.pdf

Question 1 sur 10 20:00

[{"id":21209,"question":"Quelle est la limite de (sin(x))\/x lorsque x tend vers 0 ?","option_a":"0","option_b":"1","option_c":"+∞","option_d":"n'existe pas","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le théorème des gendarmes ou la définition du nombre dérivé de sin(x) en 0 donne lim (sin(x))\/x = 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\": \"0\", \"b\": \"1\", \"c\": \"+∞\", \"d\": \"n'existe pas\"}}","_debug_options_count":4},{"id":21210,"question":"L'équivalent de ln(1+x) au voisinage de 0 est x.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équivalent de ln(1+x) au voisinage de 0 est bien x, car ln(1+x) = x - x²\/2 + o(x²).","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":21211,"question":"Quelle est la limite de (e^x - 1)\/x lorsque x tend vers 0 ?","option_a":"0","option_b":"1","option_c":"+∞","option_d":"n'existe pas","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est la définition du nombre dérivé de e^x en 0, donc la limite vaut 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\": \"0\", \"b\": \"1\", \"c\": \"+∞\", \"d\": \"n'existe pas\"}}","_debug_options_count":4},{"id":21212,"question":"L'équivalent de (1 - cos(x)) au voisinage de 0 est x²\/2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équivalent de (1 - cos(x)) au voisinage de 0 est bien x²\/2, car 1 - cos(x) = x²\/2 - x⁴\/24 + o(x⁴).","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":21213,"question":"Quelle est la limite de (x² + 3x - 4)\/(x - 1) lorsque x tend vers 1 ?","option_a":"0","option_b":"1","option_c":"5","option_d":"n'existe pas","option_e":"","option_f":"","bonne_reponse":"d","explication":"En factorisant le numérateur : (x-1)(x+4)\/(x-1) = x+4, donc la limite vaut 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"5\", \"d\": \"n'existe pas\"}}","_debug_options_count":4},{"id":21214,"question":"L'équivalent de tan(x) au voisinage de 0 est x.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équivalent de tan(x) au voisinage de 0 est bien x, car tan(x) = x + x³\/3 + o(x³).","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":21215,"question":"Quelle est la limite de (ln(x))\/(x-1) lorsque x tend vers 1 ?","option_a":"0","option_b":"1","option_c":"+∞","option_d":"n'existe pas","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est une forme indéterminée 0\/0. En utilisant l'équivalent ln(1+x) ~ x ou la règle de l'Hôpital, on trouve 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\": \"0\", \"b\": \"1\", \"c\": \"+∞\", \"d\": \"n'existe pas\"}}","_debug_options_count":4},{"id":21216,"question":"L'équivalent de (e^x - 1 - x) au voisinage de 0 est x²\/2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équivalent de (e^x - 1 - x) au voisinage de 0 est bien x²\/2, car e^x = 1 + x + x²\/2 + o(x²).","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":21217,"question":"Quelle est la limite de (x³ - 8)\/(x - 2) lorsque x tend vers 2 ?","option_a":"0","option_b":"4","option_c":"12","option_d":"n'existe pas","option_e":"","option_f":"","bonne_reponse":"c","explication":"En factorisant le numérateur : (x-2)(x²+2x+4)\/(x-2) = x²+2x+4, donc la limite vaut 12.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"4\", \"c\": \"12\", \"d\": \"n'existe pas\"}}","_debug_options_count":4},{"id":21218,"question":"L'équivalent de (1 + x)^α - 1 au voisinage de 0 est αx.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équivalent de (1 + x)^α - 1 au voisinage de 0 est bien αx, car (1+x)^α = 1 + αx + o(x).","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}]