## Calculus (3rd Edition)

Published by W. H. Freeman

# Chapter 2 - Limits - 2.3 Basic Limit Laws - Exercises - Page 59: 40

#### Answer

See the proof below.

#### Work Step by Step

Suppose that $\lim\limits_{x \to c}\frac{f(x)}{g(x)}$ exists. Now, we have $$\lim\limits_{x \to c}f(x)=\lim\limits_{x \to c}g(x)\frac{f(x)}{g(x)}=\lim\limits_{x \to c}g(x)\lim\limits_{x \to c}\frac{f(x)}{g(x)}=0\times\lim\limits_{x \to c}\frac{f(x)}{g(x)}=0\ne L$$ But this contradicts what is given. Consequently, $\lim\limits_{x \to c}\frac{f(x)}{g(x)}$ does not exist.

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