limite X rightarrow 0 {tan(pi/4+x)} 1/x

Question:

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

  1. 2

  2. $\mathrm{e}$

  3. 1

  4. $\mathrm{e}^{2}$


Correct Option: , 4

Solution:

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

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

$=e^{\lim _{x \rightarrow 9}\left(\frac{1+\tan x-1+\tan x}{x(1-\tan x)}\right)}$

$=\lim _{x \rightarrow x(1-\tan x)}$

$=\mathrm{e}^{2}$

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