Question

lim (sin(x))^(tan(x)) as x ->0+

Answer 1

\[\large\begin{align*}\lim_{x\to0^+}(\sin x)^{\tan x}&=\lim_{x\to0^+}e^{\ln(\sin x)^{\tan x}}\\\\
&=\lim_{x\to0^+}e^{\tan x\ln(\sin x)}\\\\
&=\lim_{x\to0^+}e^{\frac{\ln(\sin x)}{\cot x}}\\\\
&=e^{\lim\limits_{~~x\to0^+}\frac{\ln(\sin x)}{\cot x}}\end{align*}\]
The limit in the exponent is \(\dfrac{\infty}{\infty}\). Use L'Hopital's rule.

Answer 2

Thx.