Question

Evaluate:$y={lim}_{x\to 0}{\left(\sin x\right)}^{\tan x}$

Answer 1

$y={lim}_{x\to 0}{\left(\sin x\right)}^{\tan x}$

$\log y={lim}_{x\to 0}\log \left({\left(\sin x\right)}^{\tan x}\right)$

$={lim}_{x\to 0}\frac{\log \sin x}{\cot x}$

Applying L Hospital Rule

$\log y={lim}_{x\to 0}\frac{\frac{1}{\sin x}\times \cos x}{-{\csc }^{2}x}$

$={lim}_{x\to 0}(-\sin x\cos x)=0$

$\Rightarrow y=e^0=1$