Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 3 - The Derivative - 3.1 Limits - 3.1 Exercises - Page 136: 5


a. $\displaystyle \lim_{x\rightarrow 3}f(x)=3,\quad $ b. $\displaystyle \lim_{x\rightarrow 0}f(x)=1$

Work Step by Step

The graph consists of points (x,f(x)). When inspecting limits at x=a, approach a on the x axis and observe what happens to the y-coordinate, f(x), on the graph. A limit exists only if both one-sided limits exist, (f(x) approaches a fixed value) and are equal, ----------- (f is not defined for x=0) a. As $x$ gets closer to 3 from the left, $f(x)$ gets closer to 3. This also happens when $x$ gets closer to 3 from the right. Both one-sided limits exist, and are equal, $\displaystyle \lim_{x\rightarrow 3}f(x)=3$ b. As $x$ gets closer to $0$ from either the left or right, $f(x)$ gets closer to 1. Both one-sided limits exist, and are equal, $\displaystyle \lim_{x\rightarrow 0}f(x)=1$.
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