Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 2 - Derivatives - 2.4 Derivatives of Trigonometric Functions - 2.4 Exercises - Page 151: 49


$$\lim _{x \rightarrow \frac{\pi}{ 4}} \frac{1-\tan x}{\sin x-\cos x}=-\sqrt{2}$$

Work Step by Step

Given $$\lim _{x \rightarrow \frac{\pi}{ 4}} \frac{1-\tan x}{\sin x-\cos x}$$ so, \begin{align} \lim _{x \rightarrow \frac{\pi}{ 4}} \frac{1-\tan x}{\sin x-\cos x}&=\lim _{x \rightarrow\frac{\pi}{ 4}} \frac{\left(1-\frac{\sin x}{\cos x}\right) \cdot \cos x}{\sin x-\cos x}\\ &=\lim _{x \rightarrow \frac{\pi}{ 4}} \frac{\cos x-\sin x}{(\sin x-\cos x) \cos x}\\ &=\lim _{x \rightarrow \frac{\pi}{ 4}} \frac{-1}{\cos x}=\frac{-1}{1 / \sqrt{2}}=-\sqrt{2} \end{align}
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