Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 3 - Applications of Differentiation - 3.4 Limits at Infinity; Horizontal Asymptotes - 3.4 Exercises - Page 242: 25



Work Step by Step

Given $$\lim _{x \rightarrow \infty} \frac{x^{4}-3x^2+x}{x^{3}-x+2}$$ Then \begin{aligned} \lim _{x \rightarrow \infty} \frac{x^{4}-3x^2+x}{x^{3}-x+2} &= \lim _{x \rightarrow \infty} \frac{x^{4}/x^{3}-3x^2/x^{3}+x/x^{3}}{x^{3}/x^{3}-x/x^{3}+2/x^{3}}\\ &= \lim _{x \rightarrow \infty} \frac{x -3 /x +1/x^{2}}{1-1/x^{2}+2/x^{3}}\\ &= \frac{ \lim _{x \rightarrow \infty}x - \lim _{x \rightarrow \infty}3 /x + \lim _{x \rightarrow \infty}1/x^{2}}{ \lim _{x \rightarrow \infty}1- \lim _{x \rightarrow \infty}1/x^{2}+ \lim _{x \rightarrow \infty}2/x^{3}}\\ &= \lim _{x \rightarrow \infty}x\\ &=\infty \end{aligned}
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