Calculus: Early Transcendentals 9th Edition

Published by Cengage Learning
ISBN 10: 1337613924
ISBN 13: 978-1-33761-392-7

Chapter 2 - Section 2.4 - The Precise Definition of a Limit - 2.4 Exercises - Page 114: 36

Answer

$\lim\limits_{x \to 2} \frac{1}{x} = \frac{1}{2}$

Work Step by Step

Let $\epsilon \gt 0$ be given. Let $\delta = min\{1, \epsilon\}$ Suppose that $\vert x-2 \vert \lt \delta$ Note that $1 \lt x \lt 3$ Then: $\vert \frac{1}{x}-\frac{1}{2}\vert = \vert \frac{2-x}{2x} \vert = \vert \frac{1}{2x} \vert \cdot \vert 2-x \vert \lt (\frac{1}{2})(\delta) \leq (\frac{1}{2})(\epsilon) \lt \epsilon$ Therefore, $\lim\limits_{x \to 2} \frac{1}{x} = \frac{1}{2}$
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