Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 2 - Section 2.6 - Limits at Infinity; Horizontal Asymptotes - 2.6 Exercises - Page 139: 66

Answer

(a) These two functions seem to have the same end behavior. (b) Since the ratio of $\frac{P(x)}{Q(x)}$ approaches 1 as $~~x \to \infty~~$, $P$ and $Q$ have the same end behavior.
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Work Step by Step

(a) When we view the graphs in the viewing rectangle $[-2,2]$ by $[-2,2]$, the graphs have a general trend that is similar, but the details are somewhat different. When we view the graphs in the viewing rectangle $[-10,10]$ by $[-10,000,10,000]$, the graphs have a very similar shape. These two functions seem to have the same end behavior. (b) $\lim\limits_{x \to \infty}\frac{P(x)}{Q(x)}$ $=\lim\limits_{x \to \infty}\frac{3x^5-5x^3+2x}{3x^5}$ $=\lim\limits_{x \to \infty}\frac{3x^5/x^5-5x^3/x^5+2x/x^5}{3x^5/x^5}$ $= \frac{3}{3}$ $= 1$ Since the ratio of $\frac{P(x)}{Q(x)}$ approaches 1 as $~~x \to \infty,~~$ $P$ and $Q$ have the same end behavior.
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