The value of $\underset{x\to 0}{lim} {\left\{\tan \left(\frac{\pi }{4}+x\right)\right\}}^{1/x},$ is

Solution:

We have,

$\begin{array}{l}\underset{x\to 0}{lim} {\left\{\tan \left(\frac{\pi }{4}+x\right)\right\}}^{\frac{1}{x}}=\underset{x\to 0}{lim} {\left(\frac{1+\tan x}{1-\tan x}\right)}^{1/x}\\ =\underset{x\to 0}{lim} {\left(1+\frac{2\tan x}{1-\tan x}\right)}^{1/x}=e^{\underset{x\to 0}{lim} \frac{2\tan x}{1-\tan x}\cdot \frac{1}{x}}\\ =e^{\underset{x\to 0}{lim} \frac{2}{1-\tan x}\times \frac{\tan x}{x}} =e^2\end{array}$