Rational Functions: Horizontal Asymptotes
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Rational Functions: Horizontal Asymptotes

Horizontal Asymptotes: A horizontal asymptote of a rational function is a horizontal line (equation: y = number) such that as values of the independent variable, x, decrease without bound or increase without bound, the function values (y-values) approach (get closer and closer to) the number (from either above or below).

The next two slides illustrate the definition.


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The rational function shown graphed has horizontal asymptote: y = 2 (dashed line), since as x-values decrease without bound, y-values approach 2 from "below".

30

20

10

0

-2

2

4

6

-10

-20

-30

x decreasing without bound

x - 1 - 10 - 100 - 1000

y 0.5 1.54 1.94 1.994

y approaches 2 from "below"

Rational Functions: Horizontal Asymptotes

Slide 2


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Also note as asymptote: x-values increase without bound, y-values approach 2 from "above".

30

20

10

0

-2

2

4

6

-10

-20

-30

x increasing without bound

x 6 10 100 1000

y 4 2.86 2.06 2.006

y approaches 2 from "above"

Rational Functions: Horizontal Asymptotes

Slide 3


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Example 1: Algebraically find the horizontal asymptote of, asymptote:

Rational Functions: Horizontal Asymptotes

First, compare the degree of the polynomial in the numerator with the degree of the polynomial in the denominator. There are 3 possibilities to consider.

(1) If degree of numerator < degree of denominator, the horizontal asymptote is y = 0 (the x-axis).

(2) If degree of numerator > degree of denominator, there is no horizontal asymptote.

Slide 4


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(3) If degree of numerator = degree of denominator, the horizontal asymptote is

In the function, the degrees of the

polynomials in both the numerator and denominator are 1, so the horizontal asymptote is found using (3) above.

Rational Functions: Horizontal Asymptotes

This function’s horizontal asymptote is y = 2.

(This was the function shown graphed in the preceding slides!)

Slide 5


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Try: Algebraically find the horizontal asymptote of, the horizontal asymptote is

Try: Algebraically find the horizontal asymptote of,

Rational Functions: Horizontal Asymptotes

This horizontal asymptote is y = 0.

This horizontal asymptote is y = 1/4.

Slide 6


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Rational Functions: Horizontal Asymptotes the horizontal asymptote is

END OF PRESENTATION

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