Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 11 - Section 11.1 - Finding Limits Using Tables and Graphs - Concept and Vocabulary Check - Page 1137: 7

Answer

The complete statement is, “If $\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f\left( x \right)=L $ and $\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f\left( x \right)=M $ where $ L\ne M $, then $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)$ does not exist."

Work Step by Step

Consider the provided limit notation, $\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f\left( x \right)=L $ Here, $ f $ is any function defined on some open interval containing the number $ a $. The function f may or may not be defined at a. Hence, the notation $\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f\left( x \right)=L $ means that as $x$ gets closer to $a$ from the left, but remains unequal to $a$, the corresponding value of $ f\left( x \right)$ gets closer to L. Consider the other provided limit notation, $\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f\left( x \right)=M $; this implies that as $x$ gets closer to $a$ from the right, but remains unequal to $a$, the corresponding value of $ f\left( x \right)$ gets closer to M. Since, $\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f\left( x \right)=L $ and $\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f\left( x \right)=M $ there is no single number that the values of $ f\left( x \right)$ are close to when x is close to a. Hence we can conclude, “If $\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f\left( x \right)=L $ and $\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f\left( x \right)=M $ where $ L\ne M $, then $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)$ does not exist.
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