Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 2: Limits and Continuity - Section 2.3 - The Precise Definition of a Limit - Exercises 2.3 - Page 66: 38


See the proof below.

Work Step by Step

To prove $\lim\limits_{x \to 3}(3x-7)=2$, we let $ c=3, f(x)=3x-7, L=2$ as used in the definition of limits. For any given small value of $\epsilon\gt0$, we need to find a value of $\delta $ so that for all $ x, 0\lt |x-c|\lt\delta $, we should have $|f(x)-L|\lt\epsilon $. or $|3x-7-2|\lt\epsilon, |3(x-3)|\lt\epsilon $. Let $\delta=\epsilon/3$, for all $ x, 0\lt|x-3|\lt\delta=\epsilon/3$, we have: $|f(x)-L|=|(3x-7)-2|=3|x-3|\lt\epsilon $ which proves the limit statement.
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