## Precalculus (6th Edition) Blitzer

Published by Pearson

# Chapter 11 - Section 11.1 - Finding Limits Using Tables and Graphs - Exercise Set - Page 1139: 32

#### Answer

a) $\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,f\left( x \right)=1$, since, as x approaches 3 from the left, the value of $f\left( x \right)$ gets closer to the y-coordinate of 1. b) $\underset{x\to {{3}^{+}}}{\mathop{\lim }}\,f\left( x \right)=3$, since, as x approaches 3 from the right, the value of $f\left( x \right)$ gets closer to the y-coordinate of 3. c) $\underset{x\to 3}{\mathop{\lim }}\,f\left( x \right)$, does not exist, because both the left hand and right-hand limits are unequal at $x=3$. d) $f\left( 3 \right)=3$, because this point $\left( 3,3 \right)$ is shown by the solid dot in the provided graph. e) $\underset{x\to {{3.5}^{-}}}{\mathop{\lim }}\,f\left( x \right)=3$, since, as x approaches 3.5 from the left, the value of $f\left( x \right)$ gets closer to the y-coordinate of 3. f) $\underset{x\to {{3.5}^{+}}}{\mathop{\lim }}\,f\left( x \right)=3$, since, as x approaches 3.5 from the right, the value of $f\left( x \right)$ gets closer to the y-coordinate of 3. g) $\underset{x\to 3.5}{\mathop{\lim }}\,f\left( x \right)=3$, because the left hand and right-hand limits at $x=3.5$ are equal. h) $f\left( 3.5 \right)=3$, because this point is shown by the solid dot in the provided graph.

#### Work Step by Step

(a) Consider the provided limit $\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,f\left( x \right)$. To find $\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,f\left( x \right)$, examine the portion of the graph near $x=3$ but from the left. As x approaches 3 from the left, the value of $f\left( x \right)$ gets closer to the y-coordinate of 1. This point $\left( 3,1 \right)$ is shown by the open dot in the above graph. The point $\left( 3,1 \right)$ has a y-coordinate of 1. Thus, $\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,f\left( x \right)=1$. Hence, the value of $\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,f\left( x \right)$ is 1. (b) Consider the provided limit $\underset{x\to {{3}^{+}}}{\mathop{\lim }}\,f\left( x \right)$. To find $\underset{x\to {{3}^{+}}}{\mathop{\lim }}\,f\left( x \right)$, examine the portion of the graph near $x=3$ but from the right. As x approaches 3 from the right, the value of $f\left( x \right)$ gets closer to the y-coordinate of 3. This point $\left( 3,3 \right)$ is shown by the solid dot in the above graph. The point $\left( 3,3 \right)$ has a y-coordinate of 3. Thus, $\underset{x\to {{3}^{+}}}{\mathop{\lim }}\,f\left( x \right)=3$. Hence, the value of $\underset{x\to {{3}^{+}}}{\mathop{\lim }}\,f\left( x \right)$ is 3. (c) Consider the provided limit $\underset{x\to 3}{\mathop{\lim }}\,f\left( x \right)$. To find $\underset{x\to {{3}^{+}}}{\mathop{\lim }}\,f\left( x \right)$, examine the portion of the graph near $x=3$ but from the right. As x approaches 3 from the right, the value of $f\left( x \right)$ gets closer to the y-coordinate of 3. This point $\left( 3,3 \right)$ is shown by the solid dot in the above graph. The point $\left( 3,3 \right)$ has a y-coordinate of 3. Thus, $\underset{x\to {{3}^{+}}}{\mathop{\lim }}\,f\left( x \right)=3$. Hence, the value of $\underset{x\to {{3}^{+}}}{\mathop{\lim }}\,f\left( x \right)$ is 3. To find $\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,f\left( x \right)$, examine the portion of the graph near $x=3$ but from the left. As x approaches 3 from the left, the value of $f\left( x \right)$ gets closer to the y-coordinate of 1. This point $\left( 3,1 \right)$ is shown by the open dot in the above graph. The point $\left( 3,1 \right)$ has a y-coordinate of 1. Thus, $\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,f\left( x \right)=1$. Hence, the value of $\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,f\left( x \right)$ is 1. Since, $\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,f\left( x \right)=1$ and $\underset{x\to {{3}^{+}}}{\mathop{\lim }}\,f\left( x \right)=3$. Here, both the left-hand limit and right-hand limit at $x=3$ are unequal, Hence, $\underset{x\to 3}{\mathop{\lim }}\,f\left( x \right)$ does not exist. (d) Consider the provided function, $f\left( 3 \right)$. To find $f\left( 3 \right)$, examine the portion of the graph near $x=3$. The graph of “f” at $x=3$ is shown by the solid dot in the provided graph with coordinates $\left( 3,3 \right)$. Thus, $f\left( 3 \right)=3$. (e) Consider the provided limit $\underset{x\to {{3.5}^{-}}}{\mathop{\lim }}\,f\left( x \right)$. To find $\underset{x\to {{3.5}^{-}}}{\mathop{\lim }}\,f\left( x \right)$, examine the portion of the graph near $x=3.5$ but from the left. As x approaches 3.5 from the left, the value of $f\left( x \right)$ gets closer to the y-coordinate of 3. This point $\left( 3.5,3 \right)$ is on the parallel line shown at $y=3$ in the above graph. The point $\left( 3.5,3 \right)$ has a y-coordinate of 3. Thus, $\underset{x\to {{3.5}^{-}}}{\mathop{\lim }}\,f\left( x \right)=3$. Hence, the value of $\underset{x\to {{3.5}^{-}}}{\mathop{\lim }}\,f\left( x \right)$ is 3. (f) Consider the provided limit $\underset{x\to {{3.5}^{+}}}{\mathop{\lim }}\,f\left( x \right)$. To find $\underset{x\to {{3.5}^{+}}}{\mathop{\lim }}\,f\left( x \right)$, examine the portion of the graph near $x=3.5$ but from the right. As x approaches 3.5 from the right, the value of $f\left( x \right)$ gets closer to the y-coordinate of 3. This point $\left( 3.5,3 \right)$ is on the parallel line shown at $y=3$ in the above graph Thus $\underset{x\to {{3.5}^{+}}}{\mathop{\lim }}\,f\left( x \right)=3$. Hence, the value of $\underset{x\to {{3.5}^{+}}}{\mathop{\lim }}\,f\left( x \right)$ is 3. (g) Consider the provided limit $\underset{x\to 3.5}{\mathop{\lim }}\,f\left( x \right)$. To find $\underset{x\to {{3.5}^{+}}}{\mathop{\lim }}\,f\left( x \right)$, examine the portion of the graph near $x=3.5$ but from the right. As x approaches 3.5 from the right, the value of $f\left( x \right)$ gets closer to the y-coordinate of 3. This point $\left( 3.5,3 \right)$ is on the parallel line shown at $y=3$ in the above graph Thus $\underset{x\to {{3.5}^{+}}}{\mathop{\lim }}\,f\left( x \right)=3$. Hence, the value of $\underset{x\to {{3.5}^{+}}}{\mathop{\lim }}\,f\left( x \right)$ is 3. To find $\underset{x\to {{3.5}^{-}}}{\mathop{\lim }}\,f\left( x \right)$, examine the portion of the graph near $x=3.5$ but from the left. As x approaches 3.5 from the left, the value of $f\left( x \right)$ gets closer to the y-coordinate of 3. This point $\left( 3.5,3 \right)$ is on the parallel line shown at $y=3$ in the above graph. The point $\left( 3.5,3 \right)$ has a y-coordinate of 3. Thus, $\underset{x\to {{3.5}^{-}}}{\mathop{\lim }}\,f\left( x \right)=3$. Hence, the value of $\underset{x\to {{3.5}^{-}}}{\mathop{\lim }}\,f\left( x \right)$ is 3. Since, $\underset{x\to {{3.5}^{+}}}{\mathop{\lim }}\,f\left( x \right)$ and $\underset{x\to {{3.5}^{-}}}{\mathop{\lim }}\,f\left( x \right)=3$. Here, both the left-hand limit and right-hand limit at $x=3.5$ are equal, Hence, $\underset{x\to 3.5}{\mathop{\lim }}\,f\left( x \right)=3$. (h) To find $f\left( 3.5 \right)$, examine the portion of the graph near $x=3.5$. The graph of “f” at $x=3.5$ is shown by the solid dot in the provided graph with coordinates $\left( 3.5,3 \right)$. Thus, the function $f\left( 3.5 \right)=3$.

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