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 54: 41


$\displaystyle\lim_{x\rightarrow 0^-} \dfrac{x-\sin |x|}{x^3}=\infty$ $\displaystyle\lim_{x\rightarrow 0^+} \dfrac{x-\sin |x|}{x^3}\approx 0.167$ The limit does not exist

Work Step by Step

We have to estimate the limit: $\displaystyle\lim_{x\rightarrow \pm0} \dfrac{x-\sin |x|}{x^3}$ Graph the function: Therefore we get: $\displaystyle\lim_{x\rightarrow 0^-} \dfrac{x-\sin |x|}{x^3}=\infty$ $\displaystyle\lim_{x\rightarrow 0^+} \dfrac{x-\sin |x|}{x^3}\approx 0.167$ As the left hand limit and the right hand limit are not equal, the limit of the function in 0 does not exist. The function tends to $\infty$ when $x\rightarrow 0+$.
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