University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 2 - Section 2.4 - One-Sided Limits - Exercises - Page 85: 46


$\lim_{x\to-2^+}f(x)=\lim_{x\to2^-}f(x)=7$ $\lim_{x\to-2^-}f(x)$ cannot be decided here.

Work Step by Step

If $f$ is an even function of $x$, then $f(-x)=f(x)$. We know that $\lim_{x\to2^-}f(x)=7$. This means as $x$ approaches $2$ from the left $(x\lt2)$, the values of $f(x)$ gets closer and closer to $7$. $f$ is an even function, so its graph would be symmetric through $Oy$. All the values of $f(x)$ to the left of $2$ would be reflected in the values of $f(x)$ to the right of $-2$. So if $x$ approaches $-2$ from the right, as $-x$, $f(-x)=f(x)$, the values of $f(x)$ would also approach $7$. In other words, $\lim_{x\to-2^+}f(x)=\lim_{x\to2^-}f(x)=7$ The value of $\lim_{x\to-2^-}f(x)$, unfortunately, cannot be decided without the knowledge of $\lim_{x\to2^+}f(x)$. A graph of an even function is shown below for clarification. As you can see, $\lim_{x\to-2^+}f(x)=\lim_{x\to2^-}f(x)=7$ here, but $\lim_{x\to-2^-}f(x)$ would not exist here because the graph is not defined in the domain $(-\infty,-2)$.
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