How to Find a Horizontal Asymptote

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: no horizontal asymptote

if n/d is one, then the asymptote is the ratio of the coefficients in the natural order

for example HA is y =

if n/d is less than one, then the horizontal asymptote is the x axis.

for example HA is y = 0, the x axis

Practice – find the HA, horizontal asymptote. If there is none, write “none” else write the asymptote as an equation.

1.

2.

3.

Answers

1.

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

with a remainder

the asymptote line is 5x + 4

You try it with

answer: 3x + 2 is the slant asymptote