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 137: 61

Answer

a) The limit does not exist. b) $x = -2$ c) From parts a) and b), we can conclude that the vertical asymptote exists where the function is not defined (Here, where the denominator becomes $0$ and the function tends to $\pm\infty$)

Work Step by Step

a) For $f(x) = \frac{3x}{(x+2)^3}$ we have: $\lim\limits_{x \to -2^-}\frac{3x}{(x+2)^3} = \infty$ whereas: $\lim\limits_{x \to -2^+}\frac{3x}{(x+2)^3} = -\infty$ Since left-hand limit and right-hand limit are not equal, the limit does not exist. b) In our case there is a vertical asymptote: $x=-2$ c) The vertical asymptote is where for a given $x$-value, a $y$-value does not exist. It is where, in the graph, one can draw a straight, vertical line which does not touch the function (refer to image).
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