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Horizontal Asymptotes

Module by: Pradnya Bhawalkar, Kim Johnston. E-mail the authors

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Summary: Finding horizontal asymptotes of rational functions

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Horizontal asymptotes are horizontal lines the graph approaches.

Horizontal Asymptotes CAN be crossed.

To find horizontal asymptotes:

  • If the degree (the largest exponent) of the denominator is bigger than the degree of the numerator, the horizontal asymptote is the x-axis (y = 0).
  • If the degree of the numerator is bigger than the denominator, there is no horizontal asymptote.
  • If the degrees of the numerator and denominator are the same, the horizontal asymptote equals the leading coefficient (the coefficient of the largest exponent) of the numerator divided by the leading coefficient of the denominator

One way to remember this is the following pnemonic device: BOBO BOTN EATS DC

  • BOBO - Bigger on bottom, y=0
  • BOTN - Bigger on top, none
  • EATS DC - Exponents are the same, divide coefficients

Find the Horizontal Asymptotes of the following:

Exercise 1

fx=4xx−3 f x 4 x x 3

Solution

y=4 y 4 since the degrees are the same, divide the leading coefficients of the numerator and denominator = 41=4 4 1 4

Exercise 2

gx=5x23+x g x 5 x 2 3 x

Solution

None since the degree of the numerator is greater than the degree of the denominator.

Exercise 3

hx=-4x2x−2x+4 h x -4 x 2 x 2 x 4

Solution

y=-4 y -4

Exercise 4

gx=6x+34−x g x 6 x 3 4 x

Solution

y=0 y 0

Exercise 5

fx=3xx−12x2−5x−3 f x 3 x x 1 2 x 2 5 x 3

Solution

y=32 y 32

Exercise 6

qx=-x1−x3x2+5x−2 q x x 1 x 3 x 2 5 x 2

Solution

y=13 y 13

Exercise 7

rx=xx−82 r x x x 8 2

Solution

y=0 y 0

Exercise 8

rx=xx4−1 r x x x 4 1

Solution

y = 0

Exercise 9

gx=x−3x2+1 g x x 3 x 2 1

Solution

y=0 y 0

Exercise 10

rx=3x2+xx2+4 r x 3 x 2 x x 2 4

Solution

y=3 y 3

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