Sine over sine

Relevant wiki: L'Hopital's Rule - Basic

$\begin{aligned} L & = \lim_{x \to \frac \pi 2} \frac {{\color{#3D99F6}\sin x} - {\color{#3D99F6}\sin^{\sin x} x}}{1-{\color{#3D99F6}\sin x} + \ln ({\color{#3D99F6}\sin x} )} & \small \color{#3D99F6} \text{Let }u = \sin x \\ & = \lim_{u \to 1} \frac {{\color{#3D99F6}u}-\color{#3D99F6}u^u}{1-{\color{#3D99F6}u}+\ln {\color{#3D99F6}u}} & \small \color{#3D99F6} \text{A 0/0 cases, L'Hôpital's rule applies.} \\ & = \lim_{u \to 1} \frac {1-(\ln u+1)u^u}{-1+\frac 1u} & \small \color{#3D99F6} \text{Again a 0/0 case} \\ & = \lim_{u \to 1} \frac {-u^{u-1} - (\ln u+1)^2u^u}{-\frac 1{u^2}} & \small \color{#3D99F6} \text{Differentiating up and down w.r.t. }u \text{ again} \\ & = \boxed{2} \end{aligned}$