Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 14 - A Preview of Calculus: The Limit, Derivative, and Integral of a Function - 14.2 Algebra Techniques for Finding Limits - 14.2 Assess Your Understanding: 41

Answer

$0$

Work Step by Step

Factor each polynomial completely: $\lim _{x\rightarrow -1}\dfrac {x^{3}+2x^{2}+x}{x^{4}+x^{3}+2x+2} \\=\lim _{x\rightarrow -1}\dfrac {x\left( x^{2}+2x+1\right) }{x^{3}\left( x+1\right) +2\left( x+1\right) } \\=\lim _{x\rightarrow -1}\dfrac {x\left( x+1\right) ^{2}}{\left( x+1\right) \left( x^{3}+2\right) }$ Cancel the common factors: $\require{cancel} \\=\lim _{x\rightarrow -1}\dfrac {x\left( x+1\right) ^\cancel{{2}}}{\cancel{\left( x+1\right)} \left( x^{3}+2\right)} \\=\lim _{x\rightarrow -1}\dfrac {\left( x+1\right) \left( x\right) }{x^{3}+2} \\=\dfrac {\left( -1+1\right) \times \left( -1\right) }{\left( -1\right) ^{3}+2} \\=0$
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