How to find a Horizontal Asymptote
Given a rational function, how do you know if there’s a horizontal asymptote?
The Procedure

Check the highest power of x in the numerator – call it “n”

Check the highest power of x in the denominator – call it “d”

Now look at the ratio of n to d…in other words make a fraction with n in the
numerator and d in the denominator. You will need to note the size of the
fraction:
n/d > 1
n/d = 1
n/d < 1
if n/d is bigger than 1, then there is no horizontal asymptote…
you will find out later that there might be a kind of asymptote called an
oblique asymptote, but no horizontal one
for example:
5x 5  x 3  2
3x 2  x  1
5
 1 no horizontal asymptote
2
if n/d is one, then the asymptote is the ratio of the coefficients in the natural order
for example
5x 2  x 3  2
3x 2  x  1
2
5
 1 HA is y =
3
2
if n/d is less than one, then the horizontal asymptote is the x axis.
for example
5x 2  x 3  2
3x 5  x  1
2
 1 HA is y = 0, the x axis
5
Practice – find the HA, horizontal asymptote. If there is none, write “none” else write the
asymptote as an equation.
1.
x3
2 x  11
2.
25x 3  11x 2  5
x 2  25
3.
3x  2
9 x 2  81
Answers
1
2
1.
y
2.
none
3.
y=0
Slant Asymptotes
You graph an asymptote that is a line if the power in the numerator is exactly one higher
than the power in the denominator.
To find the line, actually do the division – long division of the denominator into the
numerator. You will get a quotient of the form: mx + b + remainder.
mx + b is the slant asymptote.
Example
f (x) 
10x 2  7 x  2
2x  3
5x  4
2 x  3 10 x  7 x  2
2
with a remainder
the asymptote line is 5x + 4
You try it with f (x) 
answer:
6x 2  7x  1
2x  1
3x + 2 is the slant asymptote