University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 2 - Section 2.4 - One-Sided Limits - Exercises - Page 84: 5


(a) $\lim_{x\to0^+}f(x)$ does not exist. (b) $\lim_{x\to0^-}f(x)=0$ (c) $\lim_{x\to0}f(x)$ does not exist.

Work Step by Step

(a) $\lim_{x\to0^+}f(x)$ does not exist since as $x$ approaches $0$ from the right, $f(x)$ fluctuates continuously from $-1$ to $1$ and vice versa. Therefore, it does not approach any single stable value, $\lim_{x\to0^+}f(x)$, hence, does not exist. (b) $\lim_{x\to0^-}f(x)=0$. As $x$ approaches $0$ from the left, $f(x)$ gets arbitrarily close to $0$. (c) Since $\lim_{x\to0^+}f(x)$ does not exist, $\lim_{x\to0}f(x)$ cannot exist.
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