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: 71

Answer

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.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.