Limit

Relevant wiki: L'Hopital's Rule - Basic

$\begin{aligned} L & = \lim_{x \to 0} \frac {\sin^{-1}x-\tan^{-1}x}{\sin x \log(1+x)} & \small \color{#3D99F6} \text{A 0/0 case, L'Hôpital's rule applies.} \\ & = \lim_{x \to 0} \frac {\frac 1{\sqrt{1-x^2}}-\frac 1{1+x^2}}{\cos x \log (1+x) + \frac {\sin x}{1+x}} & \small \color{#3D99F6} \text{Again a 0/0 case} \\ & = \frac {\frac x{(1-x^2)^{3/2} }+\frac {2x}{(1+x^2)^2}}{-\sin x\log(1+x) + \frac {\cos x}{1+x} + \frac {\cos x}{1+x}-\frac {\sin x}{(1+x)^2}} & \small \color{#3D99F6} \text{Again differentiate up and down w.r.t. }x \\ & = \frac 02 = \boxed 0 \end{aligned}$