Calculus I - Finding Vertical and Horizontal Asymptotes of Rational Functions, Study notes of Calculus

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MATH 12002 - CALCULUS I

§1.6: Vertical & Horizontal Asymptote Examples

Professor Donald L. White Department of Mathematical Sciences Kent State University

Example

Find all vertical and horizontal asymptotes for the function

f (x) = 3 x

(^2) + 2x + 7 2 x^2 − 8 x − 10

Example

Find all vertical and horizontal asymptotes for the function

f (x) = 9 −^ x

2 3 x^2 − 3 x − 18

Solution

Horizontal Asymptotes: Again f (x) is a rational function with numerator and denominator of the same degree, and so the horizontal asymptote is the quotient of the leading coefficients; that is, y = − 1 / 3.

Vertical Asymptotes: Observe that

f (x) =

9 − x^2 3 x^2 − 3 x − 18 =^

(3 + x)(3 − x) 3(x + 2)(x − 3) ,

and so the denominator is 0 when x = − 2 or x = 3. When x = − 2 , the numerator is not 0 , hence x = − 2 is a vertical asymptote. [Continued →]