The value of $\underset{\mathrm{x}\to \frac{\mathrm{\pi }}{2}}{\mathrm{Lim}}\frac{\sin \mathrm{x}-{(\sin \mathrm{x})}^{\sin \mathrm{x}}}{\sin \mathrm{x}+\log \sin \mathrm{x}}$ is

$$\begin{gathered} \underset{\mathrm{x}\to \frac{\mathrm{\pi }}{2}}{\mathrm{Lim}}\frac{\sin \mathrm{x}-{(\sin \mathrm{x})}^{\sin \mathrm{x}}}{1-\sin \mathrm{x}+\log \sin \mathrm{x}} \left(\frac{0}{0} \mathrm{form}\right) \mathrm{L}\text{'} \mathrm{Hospital} \mathrm{Rule} \\ =\underset{\mathrm{x}\to \frac{\mathrm{\pi }}{2}}{\mathrm{Lim}} \frac{\mathrm{cosx}-{(\sin \mathrm{x})}^{\sin \mathrm{x}}\left[\cos \mathrm{x} \log (\sin \mathrm{x})+\cos \mathrm{x}\right]}{-\cos \mathrm{x}+\cot \mathrm{x}} \end{gathered}$$
$=\underset{\mathrm{x}\to \frac{\mathrm{\pi }}{2}}{\mathrm{Lim}} \frac{1-{(\sin \mathrm{x})}^{\sin \mathrm{x}} (\log \sin \mathrm{x}+1)}{\mathrm{cosec} \mathrm{x}-1} \left(\frac{0}{0} \mathrm{form}\right)$$$\begin{gathered} =\underset{\mathrm{x}\to \frac{\mathrm{\pi }}{2}}{\mathrm{Lim}} \frac{-{(\sin \mathrm{x})}^{\sin \mathrm{x}} \cot \mathrm{x}-(\mathrm{logsin} \mathrm{x}+1){(\sin \mathrm{x})}^{\sin \mathrm{x}} \cos \mathrm{x}(\mathrm{logsinx}+1)}{-\mathrm{cosec} \mathrm{x} \cot \mathrm{x}} \\ =\underset{\mathrm{x}\to \frac{\mathrm{\pi }}{2}}{\mathrm{Lim}} \frac{{(\sin \mathrm{x})}^{\sin \mathrm{x}}+\left(\log { \sin \mathrm{x}+1)}^2\right(\sin \mathrm{x}{)}^{\sin \mathrm{x}}.\sin \mathrm{x}}{\mathrm{cosec} \mathrm{x}} \end{gathered}$$
$=\frac{1+1}{1}=2$