Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 2 - Limits - 2.7 Limits at Infinity - Exercises - Page 82: 18

Answer

The line $ y=0$ is the horizontal asymptote of the given function.

Work Step by Step

To find the horizontal asymptotes, we calculate the following limit \begin{align*} \lim _{x \rightarrow \pm \infty}\frac{8x^3-x^2}{7+11x-4 x^4}&= \lim _{x \rightarrow \pm\infty} \frac{x^3}{x^4}\frac{8 -x^{-1}}{7x^{-1}+11x^{-3}-4 }\\ &=\frac{8}{-4} \lim _{x \rightarrow \pm\infty} \frac{x^3}{x^4}\\ &=\frac{8}{-4} \lim _{x \rightarrow \pm\infty} x^{-1}\\ &=0.\\ \end{align*} Hence, the line $ y=0$ is the horizontal asymptote of the given function.
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