Calculus 8th Edition

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

Chapter 1 - Functions and Limits - Review - True-False Quiz: 14

Answer

False

Work Step by Step

Searching for a counterexample, we need two functions with no limit at, say, x=0, and we aim for their sum to be zero ( so the limit of the sum exists) $f(x)=\displaystyle \frac{1}{x},\quad g(x)=-f(x)=-\frac{1}{x}.$ Then, neither of the limits $\displaystyle \lim_{x\rightarrow 0}f(x),\quad \displaystyle \lim_{x\rightarrow 0}g(x)$ exist, but $\displaystyle \lim_{x\rightarrow 0}[f(x)+g(x)]=\lim_{x\rightarrow 0}0=0$, ($\displaystyle \lim_{x\rightarrow 0}[f(x)+g(x)]$ exists)
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