$\lim_{x \to 0} $ $\frac{\tan x-\sin x}{\sin^3x}$
$=\lim_{x \to 0} $ $\frac{\frac{\sin x}{\cos x}-\sin x}{\sin^3x}$
$=\lim_{x \to 0} $ $\frac{1-\cos x}{\cos x.\sin^2x}$
$=\lim_{x \to 0}$ $\frac{2\sin^2(\frac{x}{2})}{\cos x.4\sin^2(\frac{x}{2})\cos^2(\frac{x}{2})}$
$=\lim_{x \to 0}$ $\frac{1}{2\cos x.\cos^2(\frac{x}{2})}=$$\frac{1}{2.1.1}=$ $\frac{1}{2}$