Question

The value of $\left(497e^{-2}\right)\underset{x\to 0}{lim}{\left(\tan \right(\frac{\pi }{4}+x\left)\right)}^{1/x}$ is

Answer 1

We have $\underset{x\to 0}{lim}\tan (\pi /4+x{)}^{1/x}=\underset{x\to 0}{lim}{(1+\frac{2\tan x}{1-\tan x})}^{1/x}$
$=\underset{x\to 0}{lim}{(1+\frac{2\tan x}{1-\tan x})}^{\frac{1-\tan x}{2\tan x}.\frac{2\tan x}{x(1-\tan x)}}$
$=e^2(\underset{x\to 0}{lim}\frac{\tan x}{x}=1,\underset{x\to 0}{lim}(1+x{)}^{1/x}=e)$
Thus the required limit is 497.