Evaluating $\lim\limits_{x \rightarrow 0}\left(\frac{\sin x}{x}\right)^{\left(\frac{\sin x}{x-\sin x}\right)}$ gives

$e^{-1}$

$\lim\limits_{x \rightarrow 0}\left(\frac{\sin x}{x}\right)^{\frac{\sin x}{x-\sin x}}=\lim\limits_{x \rightarrow 0}(1+\frac{\sin x-x}{x})^{\frac{\sin x}{\sin x-x}(-1)}$

$=e^{-1}(as $\lim\limits_{x \rightarrow 0}(1+x)^{-1/x}=e^{-1}$)