Chapter 2 - Limits - 2.5 Evaluating Limits Algebraically - Exercises - Page 72: 33

$$\frac{1}{2}$$

\begin{align*} \lim _{\theta \rightarrow \frac{\pi}{4}}\left(\frac{1}{\tan \theta-1}-\frac{2}{\tan ^{2} \theta-1}\right)&=\lim _{\theta \rightarrow \frac{\pi}{4}} \frac{(\tan \theta+1)-2}{(\tan \theta+1)(\tan \theta-1)}\\ &=\lim _{\theta \rightarrow \frac{\pi}{4}} \frac{1}{\tan \theta+1}\\ &=\frac{1}{2} \end{align*}