A vertical asymptote occurs at
*x* = *c*
when the following are all true

1) f (c) is undefined |
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2)
f (x) = ∞ or - ∞ |
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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 |
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2) = - ∞ | |||

3) = + ∞ |

Thus,
*x* = - 1
is a vertical asymptote of
*f*
, graphed below:

Figure %:
*f* (*x*) =
has a vertical asymptote at
*x* = - 1

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:

Figure %:
*f* (*x*) =
. 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.

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:

= |