Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 2 - Limits - 2.3 Basic Limit Laws - Exercises - Page 59: 36


See the proof below.

Work Step by Step

One can write $ f(x)$ as follows $$ f(x)=\frac{f(x)g(x)}{g(x)}.$$ Then, we have $$\lim\limits_{x \to c}f(x)=\lim\limits_{x \to c}\frac{f(x)g(x)}{g(x)}.$$ We are given that the limit $\lim\limits_{x \to c}f(x)g(x)$ exists and since $\lim\limits_{x \to c}g(x)\neq 0$, then we can apply the quotient law to get the overall limit. Hence, the limit $\lim\limits_{x \to c}f(x)$ exists.
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