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Chapter 3: Problem 114

If you are given the equation of a rational function, how can you tell if the graph has a slant asymptote? If it does, how do you find its equation?

Short Answer

Expert verified
To determine the existence of a slant asymptote in a rational function, examine the degrees of the numerator and denominator. If the degree of the numerator is one more than the denominator, a slant asymptote exists. Its equation can be found by performing polynomial long division of the numerator by the denominator, and taking the quotient as the equation of the slant asymptote.

Step by step solution

01

Identify the Degree of the Functions

Check the degree of the numerator and the denominator of the rational function. If the degree of the numerator is one more than the degree of the denominator, then a slant asymptote will exist.
02

Divide the Functions

Next, perform polynomial long division or synthetic division on the rational function. Polynomial long division follows the same process as numerical long division, but with polynomials instead of numbers.
03

Find the Equation of the Slant Asymptote

The quotient from the division (ignoring the remainder) is the equation of the slant asymptote.

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