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: 60

Answer

$y = 2$

Work Step by Step

Since \begin{align*} \lim _{u \rightarrow \infty} \frac{2 u^{2}-1}{\sqrt{6+u^{4}}}&=\lim _{u \rightarrow \infty} \frac{2-1 / u^{2}}{\sqrt{6 / u^{4}+1}}\\ &=\frac{2}{\sqrt{1}}=2 \end{align*} and \begin{align*} \lim _{u \rightarrow-\infty} \frac{2 u^{2}-1}{\sqrt{6+u^{4}}}&=\lim _{u \rightarrow-\infty} \frac{2-1 / u^{2}}{\sqrt{6 / u^{4}+1}}\\ &=\frac{2}{\sqrt{1}}=2 \end{align*} We see that $f(x) $ has has a horizontal asymptote of $y = 2$.
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