Limites de fonctions

QCM Bilan Numéro 2

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)=3x3x25x+9f\left(x\right)=3x^{3}-x^{2}-5x+9 . La limite de ff en -\infty est égale à :
a.\bf{a.} ++\infty                                                                                           \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} -\infty
c.\bf{c.} 33                                                                                                    \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 00

Correction
2

limx+3x+23x+4={\mathop{\lim }\limits_{x\to +\infty }} \sqrt{3x+2} -\sqrt{3x+4}=
a.\bf{a.} 00                                                                                 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} -\infty
c.\bf{c.} ++\infty                                                                        \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} Forme indéterminée

Correction
3

Soit ff la fonction définie sur ];0[\left]-\infty;0\right[ par f(x)=2x25x+35x2+x+10f\left(x\right)=\frac{2x^{2} -5x+3}{5x^{2} +x+10} et Cf\mathscr{C_f} sa courbe représentative. Cf\mathscr{C_f} admet comme asymptote la droite d'équation :
a.\bf{a.} y=5y=-5                                                                                                   \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} y=310y=\frac{3}{10}
c.\bf{c.} y=35y=\frac{3}{5}                                                                                                    \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} y=25y=\frac{2}{5}

Correction
4

limx3x+7=\lim\limits_{x\to -\infty } \sqrt{\frac{3}{x}+7 }=
a.\bf{a.} 7\sqrt{7}                                                                                           \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 3\sqrt{3}
c.\bf{c.} 00                                                                                                \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 77

Correction
5

limx+2cos(x)+3x24={\mathop{\lim }\limits_{x\to +\infty }} \frac{2\cos \left(x\right)+3}{x^{2} } -4=
a.\bf{a.} 22                                                                                           \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 33
c.\bf{c.} 4-4                                                                                        \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 00

Correction
6

limx+sin(x)ex={\mathop{\lim }\limits_{x\to +\infty }} \sin \left(x\right)e^{-x}=
a.\bf{a.} 1-1                                                                                           \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 11
c.\bf{c.} 00                                                                                                \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} ++\infty

Correction
7

limxe2x+3={\mathop{\lim }\limits_{x\to -\infty }} e^{\frac{2}{x+3} } =
a.\bf{a.} ee                                                                                           \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 11
c.\bf{c.} 00                                                                                                \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} ++\infty

Correction
8

limx3x>3x3x29={\mathop{\lim }\limits_{\begin{array}{l} {x\to 3} \\ {x>3} \end{array}}} \frac{x-3}{x^{2} -9}=
a.\bf{a.} 00                                                                                           \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 16\frac{1}{6}
c.\bf{c.} Forme indéterminée                                          \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 39\frac{-3}{-9}

Correction
9

limx1+5x+42x+2=\lim\limits_{x\to 1^{+} } \frac{5x+4}{-2x+2} =
a.\bf{a.} ++\infty                                                                   \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} -\infty
c.\bf{c.} 42\frac{4}{2}                                                                          \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 52-\frac{5}{2}

Correction
10

limx2x+4x2+2={\mathop{\lim }\limits_{x\to -\infty }} 2x+\sqrt{4x^{2} +2}=
a.\bf{a.} ++\infty                                                                 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} -\infty
c.\bf{c.} 00                                                                          \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 2-2

Correction
11

limx+4x5x2={\mathop{\lim }\limits_{x\to +\infty }} 4x-5\sqrt{x} -2=
a.\bf{a.} -\infty                                                                                 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} ++\infty
c.\bf{c.} Forme indéterminée                                          \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 2-2

Correction
12

limx+(x32x+3)2021=\lim\limits_{x\to +\infty } \left(-x^{3} -2x+3\right)^{2021}=
a.\bf{a.} -\infty                                                                                 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} ++\infty
c.\bf{c.} Forme indéterminée                                          \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 320213^{2021}

Correction
13

limx(2x13x+5)2={\mathop{\lim }\limits_{x\to -\infty }} \left(\frac{2x-1}{3x+5} \right)^{2} =
a.\bf{a.} -\infty                                                             \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} ++\infty
c.\bf{c.} 23\frac{2}{3}                                                                   \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 49\frac{4}{9}

Correction
14

limx+3x2+7ex+4x9={\mathop{\lim }\limits_{x\to +\infty }} -3x^{2} +7e^{x} +4x-9=
a.\bf{a.} -\infty                                                             \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} ++\infty
c.\bf{c.} 3-3                                                                 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 9-9

Correction