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.