Answer
$$3$$
Work Step by Step
\begin{align*}
\lim _{\theta \rightarrow 0} \frac{\tan \theta}{\theta^{2} \cot 3 \theta}&=\lim _{\theta \rightarrow 0} \frac{\frac{\sin \theta}{\cos \theta}}{\theta^{2} \frac{\cos \theta}{\sin 3 \theta}}\\
&=\lim _{\theta \rightarrow 0} \frac{\sin \theta \sin 3 \theta}{\theta^{2} \cos \theta \cos 3 \theta}\\
&=\lim _{\theta \rightarrow 0}\left(\frac{\sin \theta}{\theta}\right)\left(\frac{\sin 3 \theta}{3 \theta}\right)\left(\frac{3}{\cos \theta \cos 3 \theta}\right)\\
&=(1)(1)\left(\frac{3}{11}\right)\\
&=3
\end{align*}
