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 - Page 308: 71


Their growth rates are comparable.

Work Step by Step

We will find the limit $\lim_{x\to\infty}\frac{\ln x^{20}}{\ln x}.$ 1) If it is equal to zero then $\ln x$ grows slower than $\ln x$; 2) If it is equal to $\infty$ then $\ln x^{20}$ grows faster than $\ln x$; 3) If it is equal to some ňon zero constant then their growth rates are comparable. We will use the logarithmic rule $\ln b^a=a\ln b$: $$\lim_{x\to\infty}\frac{\ln x^{20}}{\ln x}=\lim_{x\to\infty}\frac{20\ln x}{\ln x}=\lim_{x\to\infty}=20,$$ thus 3) is right and their growth rates are comparable.
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