Question

Evaluate the following limits:

$\underset{x\to 0}{lim}{\left(\frac{\tan x}{x}\right)}^{1/x^2}$ is $e^{1/a}$ Find a

Answer 1

Expansion of $\tan x$ is,

$\tan x=x+\frac{x^3}{3}+\cdots$

Therefore required limit
$=\underset{x\to 0}{lim}{(1+\frac{x^2}{3})}^{1/x^2}$

$\underset{x\to 0}{lim}{\left[{(1+\frac{x^2}{3})}^{3/x^2}\right]}^{1/3}=e^{1/3}$,

Here we have ignored higher order term.