Calculus AB: Applications of the Derivative
Vertical and Horizontal Asymptotes
Vertical Asymptotes
A vertical asymptote occurs at x = c when the following are all true
| 1) f (c) is undefined | |||
2) 
f (x) = ∞ or - ∞ |
|||
3) 
f (x) = ∞ or - ∞ |
Taken together, #2 and #3 mean that
f
"grows without bound" as it approaches
x = c
. This happens most often with a rational function at a value of
x
that
leads to a denominator of zero. For example, consider
f (x) = 
.
f (x)
is undefined
at
x = - 1
.
| 1) f (x) is undefined at x = - 1 | |||
2)  = - ∞ |
|||
3)  = + ∞ |
Thus, x = - 1 is a vertical asymptote of f , graphed below:
has a vertical asymptote at
x = - 1
Horizontal Asymptotes
A horizontal asymptote is a horizontal line that the graph of a function approaches,
but never touches as
x
approaches negative or positive infinity.
If
f (x) = L
or
f (x) = L
, then the line
y = L
is a horiztonal asymptote of the function
f.
For example, consider the function
f (x) = 
.
This function has a horizontal asymptote at
y = 2
on both the left and the right ends of
the graph:
. Has a horizontal asymptote at
y = 2
Note that a function may cross its horizontal asymptote near the origin, but it cannot cross it as x approaches infinity.
Intuitively, we can see that
y = 2
is a horizontal asymptote of
f
because as
x
approaches infinity,
f (x) =
behaves more and more like
f (x) = 
, which is the same as
f (x) = 2
. Although
f
behaves more and
more like this, it never actually becomes this function, so
y = 2
is approached but not
reached.
Intuition can usually lead to the right answer with these problems, but the following is a more methodical way of calculating limits at infinity.
Evaluating Limits at Infinity
In order to find horizontal asymptotes, we must evaluate limits as x approaches infinity. To evaluate the limits of rational functions at infinity, first divide each of the terms in the numerator and the denominator by the highest. For example, to evaluate
|
first divide each of the terms in the numerator and denominator by the highest power of x present in the function. In this case, that is x 3.
|
then evaluate the individual limits using the following rule: if r is a rational number greater than zero such that x r is defined for all x , then
 = 0
|
Applying this rule in this case leads to the following:
= 
|
