Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 1 - Functions and Limits - 1.7 The Precise Definition of a Limit - 1.7 Exercises: 24


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Work Step by Step

$\displaystyle \lim_{x\rightarrow a}f(x)=L$ if for every number $\epsilon > 0$ there is a number $\delta > 0$ such that the following is valid: $($if $ 0 < |x-a| < \delta$ then $|f(x)-L| < \epsilon)$ ------------- $f(x)=c$. Given any $\epsilon > 0$, we want to find a $\delta > 0$ such that $ 0 < |x-a| < \delta\ \ \Rightarrow\ \ |c-c| < \epsilon$. Since $|c-c|=0$, we can take $\delta$ to be any positive number, because the conclusion is always true, regardless of the premise. By the definition, $\displaystyle \lim_{x\rightarrow a}c=c$
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