Question

Find ${lim}_{x\to 0}{\left[\tan \right(\frac{\pi }{4}+x\left)\right]}^{1/x}$.

Answer 1

$\begin{array}{c}{lim}_{x\to 0}{\left[\tan \left(\frac{\pi }{4}+x\right)\right]}^{\frac{1}{x}}\\ =e^{{lim}_{x\to 0}\frac{1}{x}\left[\tan \left(\frac{\pi }{4}+x\right)-1\right]}\\ =e^{{lim}_{x\to 0}\frac{1}{x}\left[\frac{1+\tan x}{1-\tan x}-1\right]}\\ =e^{{lim}_{x\to 0}\frac{1}{x}\left[\frac{1+\tan x-1+\tan x}{1-\tan x}\right]}\\ =e^{{lim}_{x\to 0}\frac{1}{x}\left[\frac{2\tan x}{1-\tan x}\right]}\\ =e^2\\ Hence,solved.\end{array}$