Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 3 - Review - Concept Check - Page 319: 13

Answer

a). When the degree of the numerator is greater than that of the degree of the denominator by 1. b). Check the degrees of the numerator and denominator, and if necessary, find the leading coefficients of the variable with the highest degree in both numerator and denominator.

Work Step by Step

a). If you have a polynomial f(x) = $\frac{g(x)}{h(x)}$, and g(x) has degree 1 more than h(x), you will have f(x) = a(x) + $\frac{f(x)}{h(x)}$, where a(x) is the quotient, and r(x) is the remainder. Since r(x) is the remainder, as h(x) increases, $\frac{r(x)}{h(x)}$ approaches 0, leaving you with a degree-one line that is a(x), thus forming a slant asymptote. b). If degree of numerator > degree of denominator, end behavior is either ∞ or -∞. Simply plugging in numbers or graphing the function should tell you which one is correct. If degree of numerator < degree of denominator, the end behavior is 0. Finally, if degree of numerator = degree of denominator, the end behavior is $\frac{a}{b}$, where a and b are coefficients of the variable with highest degree in numerator and denominator, respectively.
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