•Horizontal and
Vertical
Asymptotes
Vertical Asymptote
• A term which results in zero in the
denominator causes a vertical asymptote
when the function is graphed, providing that
the function is in its lowest terms. The vertical
asymptote is found at the term which causes
the zero in the denominator.
Example 1
• Find the vertical asymptote in the following
function:
• Note that x = 5 results in a zero in the
denominator. The fraction is in simplest form.
Therefore, there is a vertical asymptote at x =
5
Example 2
•
would appear to have a vertical
asymptote at x = -1. The reason that it does not
is that the fraction may be re written as
• This simplifies to (x-1), which does not result
in a zero in the denominator. This does not
result in a vertical asymptote.
Horizontal Asymptotes
Horizontal asymptotes, when they exist, are
determined by the value approached by the
function as x gets either extremely large or
extremely small. When graphed, asymptotes
are expressed as dashed lines which the values
graphed approach but never meet
Three types of rational expressions
• There are three types of rational expressions,
as determined by the relationships of the
greatest power in the numerator and
denominator.
• If the greatest power in the denominator is
greater than the greatest power in the
numerator, there is a horizontal asymptote at
y=0
Example 3
•
The greatest power in the
numerator is 1. The greatest power in the
denominator is 2. Therefore, as |x| gets larger
and larger, there is a horizontal asymptote at
x=0
• If the greatest power in the numerator is
equal to the greatest power in the
denominator, there is a horizontal asymptote
at the ratio of the leading coefficients.
Example 4
•
• The greatest power in the numerator is one.
The greatest power in the denominator is one.
The ratio of the leading coefficients is 3/1, or
3. There is a horizontal asymptote at 3.
Proof using a table of values
•
X
F(x)
-1000
3.025
-100
3.269
-8
28
-7.001
25003
-6.999
-25000
-6
-22
1000
2.975
Proof using Algebra: Multiply
numerator and denominator by 1/x
•
• Note that the numerator simplifies to 3 – 4/x,
and the denominator simplifies to 1 + 7/x
• At extremely high and extremely low values of
x, the values of -4/x and 7/x approach zero,
resulting in a horizontal asymptote at 3/1, or 3
• If the greatest power in the numerator is one
greater than the greatest power in the
denominator, an oblique asymptote generally
results.