Limit of the given function

Question

Please help me out with this limit given below-

\begin{equation} \operatorname{Lim}_{x \rightarrow 0^+}(\sin x)^{x} \end{equation}

I tried the following substitution-

\begin{equation} x \rightarrow \frac{\pi}{2}-x \end{equation} Then \begin{equation} \operatorname{Lim}_{x \rightarrow \frac{\pi}{2}^-} \frac{(\cos x)^{\pi / 2}}{(\cos x)^{x}} \end{equation} Which is of the form zero/zero so tried to apply L'Hospital's Rule but didn't reach to the answer.

Answer 1

Try the old exponential-log trick by writing it as: $\exp(x\log(\sin(x)))$. Now carry the limit inside the exponential.