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: 8

Answer

$\delta=0.1$

Work Step by Step

We see from the graph that in order for $f(x)$ to be within $\epsilon=0.15$ of $L=7.5$, we must have $$-3.1 \lt x \lt -2.9.$$ Subtracting $-3$ from all three sides gives $$-0.1 \lt x-(-3) \lt 0.1.$$ Hence for $\delta = 0.1$, $$0 \lt |x-(-3)| \lt \delta \implies 0 \lt |f(x)-7.5| \lt \epsilon.$$
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