Calculus: Early Transcendentals 8th Edition

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

Chapter 2 - Section 2.6 - Limits at Infinity; Horizontal Asymptotes - 2.6 Exercises - Page 139: 62

Answer

The y-intercept is 0 The x-intercepts are -2, 0, and 1 $\lim\limits_{x \to \infty} x^3(x+2)^2(x-1) = \infty$ $\lim\limits_{x \to -\infty} x^3(x+2)^2(x-1) = \infty$
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Work Step by Step

$y = x^3(x+2)^2(x-1)$ When $x=0$, then $~~y = (0)^3(0+2)^2(0-1) = 0$ When $y=0$: $x^3(x+2)^2(x-1) = 0$ $x = 0, -2,1$ $\lim\limits_{x \to \infty} x^3(x+2)^2(x-1) = \infty$ This limit is the product of a large magnitude positive number, a large magnitude positive number, and a large magnitude positive number. $\lim\limits_{x \to -\infty} x^3(x+2)^2(x-1) = \infty$ This limit is the product of a large magnitude negative number, a large magnitude positive number, and a large magnitude negative number. Note that the graph does not cross the x-axis at $x = -2$ because the term $(x+2)^2$ has an even exponent.
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