Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 4 - Applications of the Derivative - 4.7 L'Hopital's Rule - 4.7 Exercises: 16

Answer

The solution is $$\lim_{x\to 0}\frac{e^x-1}{x^2+3x}=\frac{1}{3}.$$

Work Step by Step

We will apply the L'Hopital's rule to solve this limit. "LR" will stand for "Apply L'Hopital's rule". $$\lim_{x\to 0}\frac{e^x-1}{x^2+3x}=\left[\frac{0}{0}\right][\text{LH}]=\lim_{x\to 0}\frac{(e^x-1)'}{(x^2+3x)'}=\lim_{x\to0}\frac{e^x}{2x+3}=\left[\frac{e^0}{2\cdot0+3}\right]=\frac{1}{3}.$$
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