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Q. $\displaystyle\lim _{x \rightarrow 0}\left(\tan \left(\frac{\pi}{4}+x\right)\right)^{1 / x}$ is equal to

JEE MainJEE Main 2020Limits and Derivatives

Solution:

$\displaystyle\lim _{x \rightarrow 0}\left\{\tan \left(\frac{\pi}{4}+x\right)\right\}^{1 / x}$
$=e^{\displaystyle\lim _{ x \rightarrow 0 } \frac{1}{x} \left\{\tan \left(\frac{\pi}{4}+ x \right)-1\right\}}$
$=e^{\displaystyle\lim _{x \rightarrow 0}\left(\frac{1+\tan x-1+\tan x}{x(1-\tan x)}\right)}$
$=e^{\displaystyle\lim _{x \rightarrow 0} \frac{2 \tan x}{x(1-\tan x)}}$
$= e ^{2}$