Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 2 - Limits - 2.5 Evaluating Limits Algebraically - Exercises - Page 72: 28


$$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cot x}{\csc x}=0$$

Work Step by Step

Given $$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cot x}{\csc x}$$ let $$ f(x) = \frac{\cot x}{\csc x}$$ Since, we have $$ f(\frac{\pi}{2})= \frac{\cot \frac{\pi}{2}}{\csc \frac{\pi}{2}}=\frac{0}{1}=0$$ So, we get \begin{aligned} L&= \lim _{x \rightarrow \frac{\pi}{2}} \frac{\cot x}{\csc x}\\ &= \lim _{x \rightarrow \frac{\pi}{2}} \frac{\cos x}{\sin x} \frac{1}{\csc x}\\ &= \lim _{x \rightarrow \frac{\pi}{2}} \frac{\cos x}{\sin x} \sin x \\ &= \lim _{x \rightarrow\frac{\pi}{2}} \cos x\\ &=\cos \frac{\pi}{2}\\ &=0 \end{aligned}
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.