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 85: 50


To prove the limit, prove that for a $\epsilon\gt0$, there always exists a corresponding value of $\delta\gt0$ such that for all $x$: $$2\lt x\lt2+\delta\Rightarrow \Big|\frac{x-2}{|x-2|}-1\Big|\lt\epsilon$$

Work Step by Step

$$\lim_{x\to2^+}\frac{x-2}{|x-2|}=1$$ Given $\epsilon\gt0$. To prove the limit, we need to find $\delta\gt0$ such that for all $x$: $$2\lt x\lt2+\delta\Rightarrow \Big|\frac{x-2}{|x-2|}-1\Big|\lt\epsilon$$ First, we need to examine the inequality: $$\Big|\frac{x-2}{|x-2|}-1\Big|\lt\epsilon$$ $$-\epsilon\lt\frac{x-2}{|x-2|}-1\lt\epsilon$$ Since $2\lt x\lt2+\delta$, here we examine only values of $x\gt2$, where $|x-2|=x-2$. Therefore, $$-\epsilon\lt\frac{x-2}{x-2}-1\lt\epsilon$$ $$-\epsilon\lt1-1\lt\epsilon$$ $$-\epsilon\lt0\lt\epsilon$$ which is always right as we define $\epsilon\gt0$ In other words, for $x\gt2$, $\Big|\frac{x-2}{|x-2|}-1\Big|\lt\epsilon$ Take any arbitrary value of $\delta\gt0$, we would have $$2\lt x\lt2+\delta\Rightarrow\Big|\frac{x-2}{|x-2|}-1\Big|\lt\epsilon$$ According to the defintion of one-side limit, this means $\lim_{x\to2^+}\frac{x-2}{|x-2|}=1$
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