Quiz — Exercices sur les limites du fonction Logarithme.pdf

Question 1 sur 10 20:00

[{"id":24134,"question":"Quelle est la limite de ln(x) lorsque x tend vers 0 par la droite ?","option_a":"0","option_b":"+∞","option_c":"-∞","option_d":"1","option_e":"","option_f":"","bonne_reponse":"c","explication":"La fonction ln(x) tend vers -∞ lorsque x approche 0 par la droite, car ln(x) n'est pas défini pour x ≤ 0.","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\": \"+∞\", \"c\": \"-∞\", \"d\": \"1\"}}","_debug_options_count":4},{"id":24135,"question":"La limite de ln(x)\/x lorsque x tend vers +∞ est égale à 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai, car ln(x) croît plus lentement que toute fonction puissance x^a (a \u003E 0), donc ln(x)\/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":24136,"question":"Quelle méthode est adaptée pour calculer lim(x→+∞) (ln(x))^2 \/ x ?","option_a":"Règle de l'Hôpital","option_b":"Croissances comparées","option_c":"Développement limité","option_d":"Factorisation","option_e":"","option_f":"","bonne_reponse":"b","explication":"Les croissances comparées montrent que (ln(x))^2 croît moins vite que x, 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\": \"b\", \"options\": {\"a\": \"Règle de l'Hôpital\", \"b\": \"Croissances comparées\", \"c\": \"Déve","_debug_options_count":4},{"id":24137,"question":"La fonction ln(x) est-elle définie pour x = -1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux, la fonction logarithme est définie uniquement pour x \u003E 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":24138,"question":"Quelle est la limite de ln(1 + x)\/x lorsque x tend vers 0 ?","option_a":"0","option_b":"1","option_c":"+∞","option_d":"-∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette limite est égale à 1, car c'est la définition de la dérivée de ln(x) en 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\": \"-∞\"}}","_debug_options_count":4},{"id":24139,"question":"Peut-on écrire lim(x→a) ln(f(x)) = ln(lim(x→a) f(x)) pour toute fonction f ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux, cette égalité n'est valable que si lim(x→a) f(x) \u003E 0 et que la limite existe.","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":24140,"question":"Quelle est la limite de ln(x + 1) - ln(x) lorsque x tend vers +∞ ?","option_a":"0","option_b":"1","option_c":"+∞","option_d":"-∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant ln(x + 1) - ln(x) = ln(1 + 1\/x), et comme 1\/x → 0, la limite est ln(1) = 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\": \"0\", \"b\": \"1\", \"c\": \"+∞\", \"d\": \"-∞\"}}","_debug_options_count":4},{"id":24141,"question":"La fonction ln(x) est-elle continue sur ]0, +∞[ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai, la fonction logarithme est continue sur son domaine de définition ]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":24142,"question":"Quelle est la limite de x * ln(x) lorsque x tend vers 0+ ?","option_a":"0","option_b":"+∞","option_c":"-∞","option_d":"1","option_e":"","option_f":"","bonne_reponse":"c","explication":"x * ln(x) tend vers 0 par valeurs positives, car ln(x) → -∞ mais x → 0 plus rapidement.","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\": \"+∞\", \"c\": \"-∞\", \"d\": \"1\"}}","_debug_options_count":4},{"id":24143,"question":"Pour calculer lim(x→a) ln(f(x)), faut-il toujours que f soit dérivable en a ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux, il suffit que f soit continue en a et que f(a) \u003E 0 pour que la limite existe.","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}]