Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 2 - Limits - 2.2 Limits: A Numerical and Graphical Approach - Exercises - Page 54: 42


$\displaystyle\lim_{x\rightarrow 4^-} \dfrac{x+1}{x-4}=-\infty$ $\displaystyle\lim_{x\rightarrow 4^+} \dfrac{x+1}{x-4}=\infty$ The limit does not exist

Work Step by Step

We have to estimate the limit: $\displaystyle\lim_{x\rightarrow \pm4} \dfrac{x+1}{x-4}$ Graph the function: Therefore we get: $\displaystyle\lim_{x\rightarrow 4^-} \dfrac{x+1}{x-4}=-\infty$ $\displaystyle\lim_{x\rightarrow 4^+} \dfrac{x+1}{x-4}=\infty$ As the left hand limit and the right hand limit are not equal, the limit of the function in 4 does not exist. The function tends to $-\infty$ when $x\rightarrow 4-$ and to $\infty$ when $x\rightarrow 4+$. There is a vertical asymptote $x=4$.
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