Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 2 - Section 2.3 - Calculating Limits Using the Limit Laws - 2.3 Exercises - Page 104: 63

Answer

$\lim\limits_{x \to a}[f(x)~g(x)]$ may exist even though neither $\lim\limits_{x \to a}f(x)$ nor $\lim\limits_{x \to a}g(x)$ exist.

Work Step by Step

Let $f(x)$ be defined as follows: $f(x) = 0~~~$ if $x \lt 0$ $f(x) = 1~~~$ if $x \geq 0$ Let $g(x)$ be defined as follows: $g(x) = 1~~~$ if $x \lt 0$ $g(x) = 0~~~$ if $x \geq 0$ Then $\lim\limits_{x \to 0}f(x)$ does not exist and $\lim\limits_{x \to 0}g(x)$ does not exist. However, $f(x)~g(x) = 0$ for all $x$ Therefore, $\lim\limits_{x \to 0}[f(x)~g(x)]$ exists. $\lim\limits_{x \to a}[f(x)~g(x)]$ may exist even though neither $\lim\limits_{x \to a}f(x)$ nor $\lim\limits_{x \to a}g(x)$ exists.
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