University Calculus: Early Transcendentals (3rd Edition)

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

Chapter 3 - Practice Exercises - Page 204: 124

Answer

$$\lim_{x\to0}\frac{3x-\tan7x}{2x}=-2$$

Work Step by Step

$$A=\lim_{x\to0}\frac{3x-\tan7x}{2x}$$ $$A=\lim_{x\to0}\frac{3x}{2x}-\lim_{x\to0}\frac{\tan7x}{2x}$$ $$A=\lim_{x\to0}\frac{3}{2}-\frac{7}{2}\lim_{x\to0}\frac{\tan7x}{7x}$$ $$A=\frac{3}{2}-\frac{7}{2}\lim_{x\to0}\frac{\sin7x}{\cos7x\times7x}$$ $$A=\frac{3}{2}-\frac{7}{2}\Big(\lim_{x\to0}\frac{\sin7x}{7x}\times\lim_{x\to0}\frac{1}{\cos7x}\Big)$$ We have $\lim_{x\to0}\frac{\sin7x}{7x}=1$. Therefore, $$A=\frac{3}{2}-\frac{7}{2}\lim_{x\to0}\frac{1}{\cos7x}$$ $$A=\frac{3}{2}-\frac{7}{2}\times\frac{1}{\cos0}$$ $$A=\frac{3}{2}-\frac{7}{2}\times\frac{1}{1}$$ $$A=-2$$
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