_ x 0 e^ 2 | sin x | -2| x|-1 x^2 - Vidyadip

MCQ

$\lim _{x \rightarrow 0} \frac{e^{2 |\text { sin } x | \mid}-2|\sin x|-1}{x^2}$

A
is equal to -$1$
B
does not exist
C
is equal to $1$
✓
is equal to $2$

✓

Answer

Correct option: D.

is equal to $2$

d
$\lim _{x \rightarrow 0} \frac{e^{2 s i n x}-2|\sin x|-1}{x^2}$

$lim _{x \rightarrow 0} \frac{e^{2 s i n x}-2|\sin x|-1}{|\sin x|^2} \times \frac{\sin ^2 x}{x^2}$

Let $|\sin x|=t$

$\lim _{t \rightarrow 0} \frac{e^{2 t}-2 t-1}{t^2} \times \lim _{x \rightarrow 0} \frac{\sin ^2 x}{x^2}$

$=\lim _{t \rightarrow 0} \frac{2 e^{2 t}-2}{2 t} \times 1=2 \times 1=2$

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