Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 2 - Limits - 2.5 Limits at Infinity - 2.5 Exercises - Page 96: 30


$\lim _{x\rightarrow \infty }\dfrac {x^{4}+7}{x^{5}+x^{2}-x}=0;\lim _{x\rightarrow -\infty }\dfrac {x^{4}+7}{x^{5}+x^{2}-x}=0;y=0$

Work Step by Step

$\lim _{x\rightarrow \infty }\dfrac {x^{4}+7}{x^{5}+x^{2}-x}=\dfrac {\dfrac {1}{x}+\dfrac {7}{x^{5}}}{1+\dfrac {1}{x^{3}}-\dfrac {1}{x^{4}}}=\dfrac {0+0}{1+0-0}=\dfrac {0}{1}=0$ $\lim _{x\rightarrow -\infty }\dfrac {x^{4}+7}{x^{5}+x^{2}-x}=\dfrac {\dfrac {1}{x}+\dfrac {7}{x^{5}}}{1+\dfrac {1}{x^{3}}-\dfrac {1}{x^{4}}}=\dfrac {0+0}{1+0-0}=\dfrac {0}{1}=0$ so the horizontal asymptote will be $y=0$
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