Question

Solve $\underset{x\to 0}{lim}{\left(\frac{\tan x}{x}\right)}^{1/x}$

Answer 1

Expansion of $\tan x$ is,
$\tan x=x+\frac{x^3}{3}+\cdots$
Therefore required limit
$=\underset{x\to 0}{lim}{(1+\frac{x^2}{3})}^{1/x}$
$\underset{x\to 0}{lim}{\left[{(1+\frac{x^2}{3})}^{3/x^2}\right]}^{x/3}=e^0=1$,
Here we have ignored higher order term.