Quiz : Maîtrisez les limites et la continuité en Terminale Math

Question 1 sur 10 10:00

[{"id":17288,"question":"Quelle est la limite de la fonction f(x) = (x² - 4)\/(x - 2) quand x tend vers 2 ?","option_a":"0","option_b":"2","option_c":"4","option_d":"La limite n'existe pas","option_e":"","option_f":"","bonne_reponse":"B","explication":"La fonction se simplifie en f(x) = x + 2 pour x ≠ 2. Ainsi, la limite quand x tend vers 2 est 4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17289,"question":"La fonction f(x) = 1\/x est-elle continue en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La fonction n'est pas définie en x=0, donc elle ne peut pas être continue en ce point.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17290,"question":"Quelle est la limite de sin(x)\/x quand x tend vers 0 ?","option_a":"0","option_b":"1","option_c":"∞","option_d":"La limite n'existe pas","option_e":"","option_f":"","bonne_reponse":"B","explication":"C'est une limite classique en analyse. On sait que lim(x→0) sin(x)\/x = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17291,"question":"Soit f une fonction continue sur [0, 5] telle que f(0) = -2 et f(5) = 3. Peut-on affirmer qu'il existe c dans [0,5] tel que f(c) = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le théorème des valeurs intermédiaires garantit l'existence de c seulement si 0 est entre f(0) et f(5), ce qui est le cas ici.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17292,"question":"Quelle est la limite de (3x + 2)\/(2x - 1) quand x tend vers +∞ ?","option_a":"0","option_b":"3\/2","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"B","explication":"Pour les fonctions rationnelles, la limite en l'infini est le rapport des coefficients dominants : 3\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17293,"question":"La fonction f(x) = x² - 3x + 2 est-elle continue sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Les fonctions polynômes sont continues sur ℝ, donc f est continue partout.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17294,"question":"Quelle est la limite de √(x+1) - √x quand x tend vers +∞ ?","option_a":"0","option_b":"1","option_c":"∞","option_d":"La limite n'existe pas","option_e":"","option_f":"","bonne_reponse":"A","explication":"En multipliant par le conjugué, on obtient lim(x→∞) 1\/(√(x+1)+√x) = 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17295,"question":"Soit f une fonction continue sur [1, 4] avec f(1) = 5 et f(4) = -1. Peut-on affirmer que l'équation f(x) = 0 admet au moins une solution dans [1,4] ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le théorème des valeurs intermédiaires s'applique car 0 est entre f(1) et f(4).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17296,"question":"Quelle est la limite de (e^x - 1)\/x quand x tend vers 0 ?","option_a":"0","option_b":"1","option_c":"e","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"B","explication":"C'est la définition de la dérivée de e^x en 0, donc la limite est 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":17297,"question":"La fonction f(x) = |x| est-elle continue en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La fonction valeur absolue est continue partout, notamment en x=0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]