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: 61


The left and right limits are equal when $n$ is even.

Work Step by Step

Consider the case when $n$ is even (e.g. $1/x^2$), \begin{align*} \lim _{x \rightarrow 0-} \frac{1}{x^{n}}&=\infty\\ \lim _{x \rightarrow 0+} \frac{1}{x^{n}}&=\infty \end{align*} and for $n$ is odd (e.g. $1/x^3$), \begin{align*} \lim _{x \rightarrow 0-} \frac{1}{x^{n}}&=-\infty\\ \lim _{x \rightarrow 0+} \frac{1}{x^{n}}&=\infty \end{align*} We see that the left and right limits are equal for even powers of $n$, but not for odd powers.
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