Quiz — 63371fabe6e00_9-Prolongement par continuité .pdf

Question 1 sur 10 20:00

[{"id":61347,"question":"Soit la fonction f définie par f(x) = (x² - 1)\/(x - 1). Peut-on prolonger f par continuité en x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La limite de f(x) quand x tend vers 1 existe et vaut 2. On peut donc prolonger f par continuité en posant f(1) = 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":61348,"question":"Quelle est la limite de la fonction f(x) = (sin x)\/x quand x tend vers 0 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. ∞","option_d":"D. -1","option_e":"","option_f":"","bonne_reponse":"b","explication":"La limite de (sin x)\/x quand x tend vers 0 est un résultat classique en analyse, égal à 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. -1\"}}","_debug_options_count":4},{"id":61349,"question":"Une fonction discontinue en un point peut-elle être prolongeable par continuité en ce point ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Si une fonction est discontinue en un point, elle ne peut pas être prolongeable par continuité en ce point, sauf si la discontinuité est supprimable (cas du prolongement par continuité).","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":61350,"question":"Soit f(x) = (x³ - 8)\/(x - 2). Que vaut f(2) si on prolonge f par continuité en x = 2 ?","option_a":"A. 0","option_b":"B. 4","option_c":"C. 12","option_d":"D. 8","option_e":"","option_f":"","bonne_reponse":"d","explication":"En simplifiant f(x) = (x³ - 8)\/(x - 2), on obtient f(x) = x² + 2x + 4. La limite quand x tend vers 2 est donc 4 + 4 + 4 = 12. On pose f(2) = 12.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. 4\", \"c\": \"C. 12\", \"d\": \"D. 8\"}}","_debug_options_count":4},{"id":61351,"question":"La fonction f(x) = 1\/x est-elle prolongeable par continuité en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La limite de 1\/x quand x tend vers 0 n'existe pas (elle tend vers +∞ ou -∞ selon le côté). La fonction n'est donc pas prolongeable par continuité en x = 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":61352,"question":"Quelle est la valeur de la limite de f(x) = (√(x + 1) - 1)\/x quand x tend vers 0 ?","option_a":"A. 0","option_b":"B. 1\/2","option_c":"C. 1","option_d":"D. ∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant la technique de la quantité conjuguée, on trouve que la limite vaut 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. 1\/2\", \"c\": \"C. 1\", \"d\": \"D. ∞\"}}","_debug_options_count":4},{"id":61353,"question":"Un prolongement par continuité est-il toujours possible pour une fonction discontinue ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Un prolongement par continuité n'est possible que si la discontinuité est supprimable, c'est-à-dire si la limite existe et est finie en ce point.","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":61354,"question":"Soit f(x) = (x² + x - 2)\/(x - 1). Peut-on prolonger f par continuité en x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La limite de f(x) quand x tend vers 1 n'existe pas car le numérateur et le dénominateur tendent tous deux vers 0, mais la fonction n'est pas prolongeable par continuité 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":61355,"question":"Quelle est la limite de 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 un résultat classique : la limite de (e^x - 1)\/x quand x tend vers 0 est égale à 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. e\", \"d\": \"D. ∞\"}}","_debug_options_count":4},{"id":61356,"question":"Si une fonction f est prolongeable par continuité en un point a, alors f est nécessairement continue en a.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si f est prolongeable par continuité en a, alors le prolongement de f est continu en a, mais f elle-même peut ne pas être définie en a.","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}]