Answer
$$\frac{1}{2}$$
Work Step by Step
We evaluate the limit:
\begin{aligned}
\lim_{\theta \to 0}\frac{\tan \theta-\sin \theta}{\sin ^{3} \theta} &=\lim_{\theta \to 0}\frac{\frac{\sin \theta}{\cos \theta}-\sin \theta}{\sin ^{3} \theta} \\
&=\lim_{\theta \to 0}\frac{\frac{\sin \theta-\sin \theta \cos \theta}{\cos \theta}}{\sin ^{3} \theta} \\
&=\lim_{\theta \to 0}\frac{\sin \theta-\sin \theta \cos \theta}{\sin ^{3} \theta \cdot \cos \theta}\\
&=\lim_{\theta \to 0}\frac{1- \cos \theta}{\sin ^{2} \theta \cdot \cos \theta}\\
&=\lim_{\theta \to 0}\frac{1- \cos \theta}{(1-\cos ^{2} \theta) \cdot \cos \theta}\\
&=\lim_{\theta \to 0}\frac{1 }{(1+\cos \theta) \cdot \cos \theta}\\
&=\frac{1}{2}
\end{aligned}
