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

Question 1 sur 10 20:00

[{"id":54690,"question":"Quelle est la limite de la fonction f(x) = (x² + 3x - 4)\/(x - 1) lorsque x tend vers 1 ?","option_a":"A. 0","option_b":"B. 5","option_c":"C. Indéterminée","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction se simplifie en f(x) = (x + 4) pour x ≠ 1. Ainsi, la limite est 1 + 4 = 5.","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. 5\", \"c\": \"C. Indéterminée\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":54691,"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 lorsque x tend vers 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":54692,"question":"Quelle est la limite de la fonction f(x) = e^(1\/x) lorsque x tend vers 0+ ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"Lorsque x tend vers 0+, 1\/x tend vers +∞, donc e^(1\/x) tend vers +∞.","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. 0\", \"b\": \"B. 1\", \"c\": \"C. +∞\", \"d\": \"D. -∞\"}}","_debug_options_count":4},{"id":54693,"question":"L’équivalent de sin(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 sin(x) au voisinage de 0 est bien x, car sin(x) ≈ x lorsque x tend vers 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":54694,"question":"Quelle est la limite de la fonction f(x) = (ln(x))\/(x - 1) lorsque x tend vers 1 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. Indéterminée","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"En utilisant l’équivalent ln(1 + u) ≈ u pour u → 0, on obtient f(x) ≈ 1\/(x - 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\": \"c\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. Indéterminée\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":54695,"question":"L’équivalent de (1 - cos(x))\/x² au voisinage de 0 est 1\/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))\/x² au voisinage de 0 est bien 1\/2, car 1 - cos(x) ≈ x²\/2.","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":54696,"question":"Quelle est la limite de la fonction f(x) = (x^3 - 1)\/(x - 1) lorsque x tend vers 1 ?","option_a":"A. 0","option_b":"B. 3","option_c":"C. Indéterminée","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction se simplifie en f(x) = x² + x + 1 pour x ≠ 1. Ainsi, la limite est 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. Indéterminée\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":54697,"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 lorsque x tend vers 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":54698,"question":"Quelle est la limite de la fonction f(x) = (e^x - 1)\/x lorsque x tend vers 0 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. Indéterminée","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"L’équivalent de e^x - 1 au voisinage de 0 est x, donc f(x) ≈ 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\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. Indéterminée\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":54699,"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, pour tout α réel.","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}]