Calculus (3rd Edition)

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

Chapter 2 - Limits - 2.2 Limits: A Numerical and Graphical Approach - Exercises - Page 55: 45


$\displaystyle\lim_{x\rightarrow 1} \dfrac{x^5+x-2}{x^2+x-2}=2$

Work Step by Step

We have to estimate the limit: $\displaystyle\lim_{x\rightarrow 1\pm} \dfrac{x^5+x-2}{x^2+x-2}$ Graph the function: Therefore we get: $\displaystyle\lim_{x\rightarrow 1^{-}} \dfrac{x^5+x-2}{x^2+x-2}=2$ $\displaystyle\lim_{x\rightarrow 1^{+}} \dfrac{x^5+x-2}{x^2+x-2}=2$ As the left hand limit and the right hand limit are equal, we have: $\displaystyle\lim_{x\rightarrow 1} \dfrac{x^5+x-2}{x^2+x-2}=2$ There is a hole in the graph in $(1,2)$.
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