IIT-JEE 2006Question 1
For $x>0$, $\displaystyle\lim_{x\to0}\left((\sin x)^{1/x}+(1/x)^{\sin x}\right)$ is
Limits & Continuity
Solution
$(\sin x)^{1/x}=e^{\ln(\sin x)/x}\to e^{-\infty}=0$. $(1/x)^{\sin x}=e^{-\sin x\ln x}$ and $\sin x\ln x\to0$, so it tends to $1$. Limit $=0+1=1$.