Maîtrisez les limites et asymptotes — Quiz interactif pour Terminale Math

Question 1 sur 10 10:00

[{"id":8163,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 1)\/(2x² - x + 4) lorsque x tend vers +∞ ?","option_a":"A. 3\/2","option_b":"B. +∞","option_c":"C. -∞","option_d":"D. 0","option_e":"","option_f":"","bonne_reponse":"A","explication":"En divisant numérateur et dénominateur par x², on obtient (3 + 2\/x - 1\/x²)\/(2 - 1\/x + 4\/x²) → 3\/2 lorsque x → +∞.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8164,"question":"L'asymptote horizontale de la fonction f(x) = (2x + 1)\/(x - 3) est la droite d'équation y = 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La limite lorsque x → ±∞ de f(x) est 2\/1 = 2, donc l'asymptote horizontale est bien y = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8165,"question":"Quelle est la limite de la fonction f(x) = √(x² + 4x) - x lorsque x tend vers +∞ ?","option_a":"A. 0","option_b":"B. 2","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"C","explication":"En multipliant par le conjugué : (x² + 4x - x²)\/[√(x² + 4x) + x] = 4x\/[√(x² + 4x) + x] → 4x\/[x + x] = 2 lorsque x → +∞.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8166,"question":"La fonction f(x) = 1\/(x-2) admet une asymptote verticale en x = 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La limite de f(x) lorsque x → 2⁺ ou x → 2⁻ est ±∞, donc la droite x = 2 est une asymptote verticale.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8167,"question":"Quelle est la limite de la fonction f(x) = (sin x)\/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":"B","explication":"C'est une limite fondamentale : lim(x→0) (sin x)\/x = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8168,"question":"La fonction f(x) = (x² + 1)\/x admet une asymptote oblique.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"En effectuant la division euclidienne : f(x) = x + 1\/x, donc l'asymptote oblique est y = x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8169,"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. e","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"B","explication":"C'est une limite fondamentale : lim(x→0) (e^x - 1)\/x = 1 (dérivée de e^x en 0).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8170,"question":"Une asymptote horizontale peut-elle être confondue avec l'axe des abscisses ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Oui, si la limite de la fonction est 0 lorsque x → ±∞, l'asymptote horizontale est alors y = 0 (axe des abscisses).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8171,"question":"Quelle est la limite de la fonction f(x) = ln(x)\/x lorsque x tend vers +∞ ?","option_a":"A. 0","option_b":"B. +∞","option_c":"C. 1","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"A","explication":"Par croissance comparée, ln(x) croît moins vite que x, donc lim(x→+∞) ln(x)\/x = 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8172,"question":"La fonction f(x) = (x³ + 2x)\/(x² + 1) admet une asymptote oblique.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"En effectuant la division euclidienne : f(x) = x + (2x - x)\/(x² + 1), donc l'asymptote oblique est y = x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]