Calculus (3rd Edition)

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

Chapter 2 - Limits - Chapter Review Exercises - Page 95: 59

Answer

$ f(x)$ does not have any horizontal asymptotes.

Work Step by Step

Since \begin{align*} \lim _{x \rightarrow \infty} \frac{x^{2}-3 x^{4}}{x-1}&=\lim _{x \rightarrow \infty} \frac{1 / x^{2}-3}{1 / x^{3}-1 / x^{4}}\\ &=-\infty \end{align*} and \begin{align*} \lim _{x \rightarrow-\infty} \frac{x^{2}-3 x^{4}}{x-1}&=\lim _{x \rightarrow-\infty} \frac{1 / x^{2}-3}{1 / x^{3}-1 / x^{4}}\\ &=\infty \end{align*} We see that both limits are infinite and do not match, so the function $ f(x)$ does not have any horizontal asymptotes.
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