Chapter 8, Problem 16RE

To determine

To find: The limit $\underset{\theta \to \pi /2}{\lim }\left(\theta -\frac{\pi }{2}\right)\sec \theta$ .

Answer to Problem 16RE

  $\underset{\theta \to \pi /2}{\lim }\left(\theta -\frac{\pi }{2}\right)\sec \theta =-1$

Explanation of Solution

Given information:

The limit, $\underset{\theta \to \pi /2}{\lim }\left(\theta -\frac{\pi }{2}\right)\sec \theta$ .

Calculation:

Evaluate the limit as follows:

  $\begin{array}{c}\underset{\theta \to \pi /2}{\lim }\left(\theta -\frac{\pi }{2}\right)\sec \theta =\underset{\theta \to \frac{\pi }{2}}{\lim }\left(\frac{-\frac{\pi }{2}+\theta }{\frac{1}{\sec \left(\theta \right)}}\right)\\ =\underset{\theta \to \frac{\pi }{2}}{\lim }\left(\frac{1}{-\sin \left(\theta \right)}\right)\left(\text{L'Hospital's rule}\right)\\ =\frac{1}{-1}\\ =-1\end{array}$

Conclusion:

The value of the limit is $\underset{\theta \to \pi /2}{\lim }\left(\theta -\frac{\pi }{2}\right)\sec \theta =-1$ .