University Calculus: Early Transcendentals (3rd Edition)

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

Chapter 2 - Section 2.3 - The Precise Definition of a Limit - Exercises - Page 75: 2

Answer

We take $\delta=1$ here.
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Work Step by Step

$$a=1 \hspace{1cm} b=7\hspace{1cm}c=2$$ Here we need to find $\delta\gt0$ such that for all $x$, $0\lt |x-2|\lt\delta$ then $1\lt x\lt 7$ The sketch is shown below. The distance from $c=2$ to the nearer endpoint of $(1,7)$ is $1$. So if we take $\delta=1$ or any smaller positive value, then $0\lt |x-2|\lt1$, and $x$ will only range in $(1,3)$, as shown by the red lines in the sketch, thereby $x$ is always placed between $1$ and $7$ as requested. In other words, $$0\lt |x-2|\lt1\hspace{2cm}1\lt x\lt 7$$
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