Answer
$$\lim_{\theta\to0}\frac{\tan\theta}{\theta^2\cot3\theta}=3$$
Work Step by Step
$$A=\lim_{\theta\to0}\frac{\tan\theta}{\theta^2\cot3\theta}=\lim_{\theta\to0}\frac{\frac{\sin\theta}{\cos\theta}}{\theta^2\frac{\cos3\theta}{\sin3\theta}}=\lim_{\theta\to0}\frac{\sin\theta\sin3\theta}{\theta^2\cos\theta\cos3\theta}$$
$$A=\lim_{\theta\to0}\frac{\sin\theta}{\theta}\times\lim_{\theta\to0}\frac{\sin3\theta}{\theta}\times\lim_{\theta\to0}\frac{1}{\cos\theta\cos3\theta}$$
$$A=X\times Y\times Z$$
1) Solve $X$: $$X=\lim_{\theta\to0}\frac{\sin\theta}{\theta}$$
Appy Theorem 7: $$X=1$$
2) Solve $Y$: $$Y=\lim_{\theta\to0}\frac{\sin3\theta}{\theta}$$
Multiply both numerator and denominator by $3$ $$Y=3\lim_{\theta\to0}\frac{\sin3\theta}{3\theta}$$
Apply Theorem 7 with $3\theta$: $$Y=3\times1=3$$
3) Solve $Z$: $$Z=\lim_{\theta\to0}\frac{1}{\cos\theta\cos3\theta}=\frac{1}{\cos0\cos0}=\frac{1}{1\times1}=1$$
Therefore, $$A=X\times Y\times Z=1\times3\times1=3$$
