Find the value of the following limit $$\lim_{x \to 0} x^{\sin x}$$
My attempt:
$$\begin{align*}
\text{Let: }\\
y &= \lim_{x \to 0} \Bigl [ x^{\sin x} \Bigr ]\\[1em]
\text{Then,}\\
\log y &= \lim_{x \to 0} \Bigl [ \sin x \cdot \log x \Bigr ] & \Bigl \{ 0 \cdot \infty \text{ form}\\[1em]
&= \lim_{x \to 0} \Bigl [ \frac{\sin x}{x} \cdot x \log x \Bigr ] & \Bigl \{ \text{multiply by } \frac xx \\[1em]
&= \lim_{x \to 0} \frac{\sin x}{x} \quad\times\quad \lim_{x \to 0} x \cdot \log x
& \Bigl \{\substack{\text{Product of limits $=$ Limit of products}\\\text{provided the limits exist}}\\[1em]
&= 1 \cdot \left ( \lim_{x \to 0} x \cdot \log x \right )
\end{align*}$$
Can I apply a $0 \cdot \infty$ form now? How do I proceed further?
