Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 2 - Review - Concept Check: 5

Answer

(a)$$\lim_{x\to a^+} f(x)=+\infty,\text{ or }\lim_{x\to a^+}f(x)=-\infty,\text{ or }\lim_{x\to a^-}f(x)=+\infty,\text{ or }\lim_{x\to a^-}f(x)=-\infty.$$ (b)$$\lim_{x\to+\infty} f(x)= L,\text{ or }\lim_{x\to-\infty}f(x)=L.$$

Work Step by Step

(a) By definition, the line $x=a$ is a vertical asymptote of the function $f(x)$ if $$\lim_{x\to a^+} f(x)=+\infty,\text{ or }\lim_{x\to a^+}f(x)=-\infty,\text{ or }\lim_{x\to a^-}f(x)=+\infty,\text{ or }\lim_{x\to a^-}f(x)=-\infty.$$ Thus we have $4$ possibilities: $f(x)$ goes to $+\infty$ when $x\to a^+$ ($x$ approaches $a$ from the right side), $f(x)$ goes to $-\infty$ when $x\to a^+$ ($x$ approaches $a$ from the right side), $f(x)$ goes to $+\infty$ when $x\to a^-$ ($x$ approaches $a$ from the left side), $f(x)$ goes to $-\infty$ when $x\to a^-$ ($x$ approaches $a$ from the left side) and these possibilities are on the left portion of the graph below. (b) The line $y=L$ is a horizontal asymptote to the curve $y=f(x)$ if $$\lim_{x\to+\infty} f(x)= L,\text{ or }\lim_{x\to-\infty}f(x)=L.$$ Different possibilities are on the right portion of the figure below.
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