Evaluating limlimits_x → 0≤ft((sin x)/x)^≤ft((sin x)/(x-sin x)) gives

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

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