Asymptotes

Objective:

-Be able to find vertical and horizontal asymptotes

Asymptotes:

• An asymptote is a line or curve that describes the behavior of a graph.

• The two kinds of asymptotes we will be working with are vertical and horizontal.

Types of Asymptotes

Vertical Horizontal

x = a y = b

2

2

4

2

xy

x

1

1y

x

Finding a Vertical Asymptote

• Vertical asymptotes have an equation of x = a, where a is a value that makes only the denominator equal zero

1

1y

x

( 3)( 1)

xy

x x

Examples

1) Find the vertical asymptote(s) of

( 3)( 4)( )

( 2)( 1)

x xf x

x x

Examples

2) Find the vertical asymptote(s) of 2

2

( 1)( )

( 4)

xf x

x

Examples

3) Find the vertical asymptote(s) of 2

2

3 2( )

9

x xf x

x

Examples

4) Find the vertical asymptote(s) of 2

2

5 6( )

9

x xg x

x

Finding a Horizontal Asymptote

• Horizontal asymptotes of a function, f(x), have an equation of y = b, where lim ( ) lim ( )

x xf x b or f x b

12yx

2

2

4

2

xy

x

Examples

5) Find the horizontal asymptote(s) of 2

2

3( )

1

xf x

x

Examples

6) Find the horizontal asymptote(s) of 2

2

2 4 1( )

4 3 2

x xg x

x x

Examples

7) Find the horizontal asymptote(s) of

2

4( )

2 1

xg x

x

Examples

8) Find the horizontal asymptote(s) of

2

3( )

1

xg x

x

Examples

9) Find the horizontal asymptote(s) of

2

2( ) 3

3 1

xg x

x

In summary:

• We will be working with two types of asymptotes: vertical and horizontal

• To find a vertical asymptote, find where only the denominator is equal to zero

• To find a horizontal asymptote, findlim ( ) lim ( )x x

f x b or f x b