_ x 0 ( ( 4 +x ) )^ 1 x is equal to - Vidyadip

MCQ

$\lim \limits_{x \rightarrow 0}\left(\tan \left(\frac{\pi}{4}+x\right)\right)^{\frac{1}{x}}$ is equal to

A
$2$
B
$e$
C
$1$
✓
$e^{2}$

✓

Answer

Correct option: D.

$e^{2}$

d
$\lim \limits_{x \rightarrow 0}\left\{\tan \left(\frac{\pi}{4}+x\right)\right\}^{1 / x}$

$=\lim \limits_{x \rightarrow 0} \frac{1}{x}\left\{\tan \left(\frac{\pi}{4}+x\right)-1\right\}$

$=\lim \limits_{x \rightarrow 0}\left(\frac{1+\tan x-1+\tan x}{x(1-\tan x)}\right)$

$=e^{\lim \limits_{x \rightarrow 0} \frac{2 \tan x}{x(1-\tan x)}}$

$=e^{2}$

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