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: 27


$$\lim _{x \rightarrow 0} \frac{\cot x}{\csc x}=1$$

Work Step by Step

Given $$\lim _{x \rightarrow 0} \frac{\cot x}{\csc x}$$ let $$ f(x) = \frac{\cot x}{\csc x}$$ Since, we have $$ f(0)= \frac{\cot 0}{\csc 0}=\frac{0}{0}$$ So, transform algebraically and cancel, we get \begin{aligned} L&= \lim _{x \rightarrow 0} \frac{\cot x}{\csc x}\\ &= \lim _{x \rightarrow 0} \frac{\cos x}{\sin x} \frac{1}{\csc x}\\ &= \lim _{x \rightarrow 0} \frac{\cos x}{\sin x} \sin x \\ &= \lim _{x \rightarrow 0} \cos x\\ &=\cos 0\\ &=1 \end{aligned}
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