Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 1 - Functions and Limits - 1.5 The Limit of a Function - 1.5 Exercises: 4

Answer

(a) $\lim\limits_{x \to 2^-}$f(x) = 3 (b) $\lim\limits_{x \to 2^+}$f(x) = 1 (c) does not exist because $\lim\limits_{x \to 2^-}$f(x)$\ne$$\lim\limits_{x \to 2^+}$f(x) (d) f(2)=3 (e) $\lim\limits_{x \to 4}$f(x) = 4 (f) f(4) does not exist because there is a hole

Work Step by Step

(a) As x approaches 2 from the left hand side, y goes to 3. (b) As x approaches 2 from the right hand side, y goes to 1. (c) The question is asking for y when x approaches 2 from both the left and right hand side. Because $\lim\limits_{x \to 2^-}$f(x)$\ne$$\lim\limits_{x \to 2^+}$f(x) from the answers to (a) and (b), the answer does not exist. There is only an answer when both sides go to the same y-value. (d) There is a point at (2,3) based on the graph so f(2)=3. (e) As x approaches 4 from both the left and right hand side, y goes to 4. (f) f(4) does not exist because there is a hole when x=4. As such, a y-value does not exist hence f(4) also doesn't exist.
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