TITLE: Evaluating $\lim_{x \to 0+} \left[ \sin(x)^{\frac{1}{x}}+\left(\frac{1}{x}\right)^{\sin(x)}\right] $?
QUESTION [5 upvotes]: For $x>0$, $$\lim_{x \rightarrow 0} \left[ \sin(x)^{\frac{1}{x}}+\left(\frac{1}{x}\right)^{\sin(x)}\right]
$$These are two forms of $0^\infty$ and $\infty^0$. I know these are to be evaluated separately and then added. But how do I start?

REPLY [4 votes]: Note that $0^{\infty}$ is not an indeterminate form. Then
$$\lim_{x \rightarrow 0^+} \left[ \sin(x)^{\frac{1}{x}}+\left(\frac{1}{x}\right)^{\sin(x)}\right]=\lim_{x \rightarrow 0^+} \left(\frac{1}{x}\right)^{\sin(x)}=\lim_{y \rightarrow +\infty}y^{\sin(1/y)}=\lim_{y \rightarrow +\infty}(y^{1/y})^{y\sin(1/y)}=1$$
The other limit to $0^-$is complex infinity.