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: 15



Work Step by Step

Given $$ \lim _{x\to \infty } \frac{\left(2x^2+1\right)^2}{\left(x-1\right)^2\left(x^2+x\right)} $$ \begin{aligned} \lim _{x\to \infty } \frac{\left(2x^2+1\right)^2}{\left(x-1\right)^2\left(x^2+x\right)} &= \lim _{x\to \infty }\frac{4x^4+4x^2+1}{x^4-x^3-x^2+x}\\ &= \lim _{x\to \infty }\frac{\frac{4x^4}{x^4}+\frac{4x^2}{x^4}+\frac{1}{x^4}}{\frac{x^4}{x^4}-\frac{x^3}{x^4}-\frac{x^2}{x^4}+\frac{x}{x^4}}\\ &=\lim _{x\to \infty }\frac{4+\frac{4}{x^2}+\frac{1}{x^4}}{1-\frac{1}{x }-\frac{1}{x^2}+\frac{1}{x^3}}\\ &=\lim _{x\rightarrow \infty}\frac{4+0+0}{1-0-0+0}\\ &=4 \end{aligned}
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