Calculus 8th Edition

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

Chapter 1 - Functions and Limits - Review - Exercises: 41

Answer

please see step-by-step

Work Step by Step

By definition , $\displaystyle \lim_{x\rightarrow 2}(14-5x)=4$ is true if for any $\epsilon > 0$, there is a $\delta > 0$ such that if $0 < |x-2| < \delta $, then $|(14-5x)-4| < \epsilon$. Analyzing the last expression, $|(14-5x)-4| < \epsilon$ $|-5x+10| < \epsilon$ $|-5||x-2| < \epsilon$ $|x-2| < \displaystyle \frac{\epsilon}{5}$. So, for any given $\epsilon$, if we choose $\displaystyle \delta=\frac{\epsilon}{5}$,we will have $ 0 < |x-2| < \delta \Rightarrow |(14-5x)-4|$
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