Question:
$\lim _{x \rightarrow 0}\left(\tan \left(\frac{\pi}{4}+x\right)\right)^{1 / x}$ is equal to :
Correct Option: , 4
Solution:
$\lim _{x \rightarrow 0}\left\{\tan \left(\frac{\pi}{4}+x\right)\right\}^{1 / x}$
$=\lim _{x \rightarrow 0} \frac{1}{x}\left\{\tan \left(\frac{x}{4}+x\right)-1\right\}$
$=e^{\lim _{x \rightarrow 9}\left(\frac{1+\tan x-1+\tan x}{x(1-\tan x)}\right)}$
$=\lim _{x \rightarrow x(1-\tan x)}$
$=\mathrm{e}^{2}$