displaystyle limx → 0 (ex2 - cos x/ sin2x) is equal to :

Question

displaystyle limx → 0 (ex2 - cos x/ sin2x) is equal to : (A) 3 (B) (3/2) (C) (5/4) (D) 2

Tardigrade Admin · Accepted Answer

displaystyle lim x arrow 0 (ex2- cos x/ sin 2 x) ((0/0) form) = displaystyle lim x arrow 0 (2 x ex2+ sin x/ sin 2 x) ((0/0) form) = displaystyle lim x arrow 0 (2 x(2 x ex2)+2 ex2+ cos x/2 cos 2 x)=(3/2) (Using L-Hospital Rule)

Q. $x \to 0 lim \frac{e ^{x^{2}} - cos x}{sin ^{2} x}$ is equal to :

$3$

$\frac{3}{2}$

$\frac{5}{4}$

$2$

$x \to 0 lim \frac{e ^{x^{2}} - cos x}{sin ^{2} x}$ ( $\frac{0}{0}$ form)
$= x \to 0 lim \frac{2 x e ^{x^{2}} + sin x}{sin 2 x}$ ( $\frac{0}{0}$ form)
$= x \to 0 lim \frac{2 x ( 2 x e ^{x^{2}} ) + 2 e ^{x^{2}} + cos x}{2 cos 2 x} = \frac{3}{2}$
(Using L-Hospital Rule)

Q. x→0lim​sin2xex2−cosx​ is equal to :

3

23​

45​

2

x→0lim​sin2xex2−cosx​ (00​ form) =x→0lim​sin2x2xex2+sinx​ (00​ form) =x→0lim​2cos2x2x(2xex2)+2ex2+cosx​=23​ (Using L-Hospital Rule)

Q. $x \to 0 lim \frac{e ^{x^{2}} - cos x}{sin ^{2} x}$ is equal to :

$3$

$\frac{3}{2}$

$\frac{5}{4}$

$2$

$x \to 0 lim \frac{e ^{x^{2}} - cos x}{sin ^{2} x}$ ( $\frac{0}{0}$ form)
$= x \to 0 lim \frac{2 x e ^{x^{2}} + sin x}{sin 2 x}$ ( $\frac{0}{0}$ form)
$= x \to 0 lim \frac{2 x ( 2 x e ^{x^{2}} ) + 2 e ^{x^{2}} + cos x}{2 cos 2 x} = \frac{3}{2}$
(Using L-Hospital Rule)