Chapter 9, Problem 17RE

To determine

To Find the limit.

Answer to Problem 17RE

The solution is $-1$

Explanation of Solution

Given information:

Given question is,

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

CALCULATION:

The limit is obtained as:

  $\begin{array}{l}\underset{\theta \to \frac{\pi }{2}}{\lim }\left(\theta -\frac{\pi }{2}\right)\sec \theta \\ =\underset{\theta \to \frac{\pi }{2}}{\lim }\frac{\left(\theta -\frac{\pi }{2}\right)}{\cos \pi }\left[\text{since, }\sec \theta =\frac{1}{\cos \theta }\right]\\ \underset{\theta \to \frac{\pi }{2}}{\lim }\frac{\frac{d}{d\theta }\left(\theta -\frac{\pi }{2}\right)}{\frac{d}{d\theta }\cos \theta }\\ \underset{\theta \to \frac{\pi }{2}}{\lim }\left(\frac{1-0}{-\sin \theta }\right)\left[\text{since, }\frac{d}{dx}\cos x=-\sin x,\frac{d}{dx}x=1,\frac{d}{dx}c=0\right]\\ =\frac{1}{-\sin \frac{\pi }{2}}\left[\text{since, }\sin \frac{\pi }{2}=1\right]\\ =\frac{1}{-1}=-1\end{array}$

Hence,

The solution is $-1$ .