Answer
$$-31.98$$
Work Step by Step
Given $$\lim _{\theta \rightarrow 0} \frac{\sin ^{2} 4 \theta}{\cos \theta-1}$$ Consider $$ f(\theta)= \frac{\sin ^{2} 4 \theta}{\cos \theta-1}$$ From the following figure, we can observe that \begin{align*} \lim _{\theta\rightarrow 0^+} f( \theta)&=-31.98\\ \lim _{\theta \rightarrow 0^-} f( \theta)&=-31.98 \end{align*} Then $$\lim _{\theta \rightarrow 0} \frac{\sin ^{2} 4 \theta}{\cos \theta-1}=-31.98$$
