Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 1 - Limits and Their Properties - 1.2 Exercises: 24

Answer

$(a)$ $f(-2)$ does not exist. $(b)$ $\lim_{x\to-2}f(x)$ does not exist. $(c)$ $f(0)=4$ $(d)$ $\lim_{x\to0}f(x)$ does not exist. $(e)$ $f(2)$ does not exist. $(f)$ $\lim_{x\to2}f(x)=\dfrac{1}{2}$ $(g)$ $f(4)=2$ $(h)$ $\lim_{x\to4}f(x)=\infty$
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Work Step by Step

The graph of the function $f$ is shown below: $(a)$ $f(-2)$ From the graph, it can be seen that $f$ is not defined for $-2$ because $x=-2$ represents a vertical asymptote. $f(-2)$ does not exist. $(b)$ $\lim_{x\to-2}f(x)$ From the graph, it can be seen that $f$ goes to $-\infty$ going to $-2$ from the left and to $\infty$ going to $-2$ from the right. Since the limits from the left and right are not the same, $\lim_{x\to-2}f(x)$ does not exist. $(c)$ $f(0)$ From the graph, it can be easily seen that $f(0)=4$, because of the two dots present at $x=0$, the one that is filled falls on $4$. $(d)$ $\lim_{x\to0}f(x)$ From the graph, it can be seen that $f$ goes to $\dfrac{1}{2}$ going to $0$ from the left and to $4$ going to $0$ from the right. Since the limits from the left and right are not the same, $\lim_{x\to0}f(x)$ does not exist. $(e)$ $f(2)$ From the graph, it can be seen that $f$ is not defined for $2$ because the dot is not filled at that point. $f(2)$ does not exist. $(f)$ $\lim_{x\to2}f(x)$ From the graph, it can be seen that $f$ goes to $\dfrac{1}{2}$ going to $2$ from the left and to $\dfrac{1}{2}$ going to $2$ from the right. Since the limits from the left and right are the same, $\lim_{x\to2}f(x)=\dfrac{1}{2}$. $(g)$ $f(4)$ From the graph, it can be seen that $f(4)=2$, because the dot is filled at that point. $(h)$ $\lim_{x\to4}f(x)$ From the graph, it can be seen that $f$ goes to $\infty$ going to $4$ from the left and to $\infty$ going to $4$ from the right. Since the limits from the left and right are the same, $\lim_{x\to4}f(x)=\infty$
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