Limites de fonctions

QCM2

Exercice 1

Cet exercice est un questionnaire à choix multiples (QCM). Pour chacune des questions ci-dessous, une seule des réponses est exacte. Pour chaque question, vous devez bien sur justifier.
1

Soit ff la fonction définie sur R\mathbb{R} par f(x)=x2x1f\left(x\right)=-x^{2}-x-1 . La limite de ff en -\infty est égale à :
a.\bf{a.} -\infty                                                                                                   \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} ++\infty

c.\bf{c.} 1-1                                                                                                    \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 00

Correction
2

Soit ff la fonction définie sur [0;+[\left[0;+\infty\right[ par f(x)=6x2+72x2+3f\left(x\right)=\frac{-6x^{2}+7}{2x^{2}+3} et Cf\mathscr{C_f} sa courbe représentative. Cf\mathscr{C_f} admet comme asymptote la droite d'équation :
a.\bf{a.} y=6y=-6                                                                                                   \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} y=0y=0

c.\bf{c.} y=3y=-3                                                                                                    \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} y=73y=\frac{7}{3}

Correction
3

limx+5ex2x4={\mathop{\lim }\limits_{x\to +\infty }} \frac{-5e^{x} }{2x^{4} } =
a.\bf{a.} -\infty                                                                                                   \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 00

c.\bf{c.} ++\infty                                                                                                    \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 11

Correction
4

limx+25x+3x=\lim\limits_{x\to +\infty } \sqrt{\frac{25x+3}{x} }=
a.\bf{a.} 33                                                                                                   \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 00

c.\bf{c.} 2525                                                                                                \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 55

Correction
5

limx4x<47x12x+8=\lim\limits_{ \begin{array}{l} {x\to 4} \\ {x<4} \end{array} }\frac{-7x-1}{-2x+8}=
a.\bf{a.} 72\frac{7}{2}                                                                                                   \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} ++\infty

c.\bf{c.} 18-\frac{1}{8}                                                                                                \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} -\infty

Correction
6

limx+7+sin(x)x+3=\lim\limits_{x\to +\infty }\frac{7+\sin \left(x\right)}{x+3}=
a.\bf{a.} 77                                                                                                   \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 11

c.\bf{c.} 13\frac{1}{3}                                                                                                \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 00

Correction
7

limxcos(x)+3x=\lim\limits_{x\to -\infty }\cos \left(x\right)+3x=
a.\bf{a.} 00                                                                                                   \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} ++\infty

c.\bf{c.} 1-1                                                                                              \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} -\infty

Correction
8

limx1x>15x21x2=\lim\limits_{ \begin{array}{l} {x\to 1} \\ {x>1} \end{array} }\frac{5x-2}{1-x^{2}}=
a.\bf{a.} 55                                                                                               \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} ++\infty

c.\bf{c.} 22                                                                                                \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} -\infty

Correction
9

limx(4x25x)5=\lim\limits_{x\to -\infty } \left(\frac{-4x^{2}-5}{x}\right)^{5} =
a.\bf{a.} -\infty                                                                                                   \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 4-4

c.\bf{c.} ++\infty                                                                                                    \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 5-5

Correction
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