University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 3 - Section 3.5 - Derivatives of Trigonometric Functions - Exercises - Page 152: 51


$$\lim_{x\to0}\sec\Big[e^x+\pi\tan\Big(\frac{\pi}{4\sec x}\Big)-1\Big]=-1$$

Work Step by Step

$$A=\lim_{x\to0}\sec\Big[e^x+\pi\tan\Big(\frac{\pi}{4\sec x}\Big)-1\Big]$$ $$A=\lim_{x\to0}\sec\Big[e^x+\pi\tan\Big(\frac{\pi\cos x}{4}\Big)-1\Big]$$ $$A=\sec\Big[e^0+\pi\tan\Big(\frac{\pi\cos0}{4}\Big)-1\Big]$$ $$A=\sec\Big[1+\pi\tan\Big(\frac{\pi}{4}\Big)-1\Big]$$ $$A=\sec\Big[\pi\times1\Big]$$ $$A=\sec\pi=\frac{1}{\cos\pi}=\frac{1}{-1}=-1$$
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