An asymptote is a line that a graph approaches without touching.

If a graph has a horizontal asymptote of
*y* = *k*
, then part of the graph
approaches the line
*y* = *k*
without touching it--
*y*
is almost equal to
*k*
, but
*y*
is never exactly equal to
*k*
. The following graph has a horizontal
asymptote of
*y* = 3
:

Horizontal Asymptote
*y* = 3

Horizontal Asymptote
*y* = 0

If a graph has a vertical asymptote of
*x* = *h*
, then part of the graph
approaches the line
*x* = *h*
without touching it--
*x*
is almost equal to
*h*
,
but
*x*
is never exactly equal to
*h*
. The following graph has a vertical
asymptote of
*x* = 3
:

Vertical Asymptote
*x* = 3

One reason vertical asymptotes occur is due to a zero in the denominator of a
rational function. For example, if
*f* (*x*) =
, then
*x*
cannot
equal 5, but
*x*
can equal values very close to 5 (4.99, for example). The
graph of
*f* (*x*) =
looks like:

Similarly, horizontal asymptotes occur because
*y*
can come close to a value,
but can never equal that value. In the previous graph, there is no value of
*x*
for which
*y* = 0
(
≠ 0
), but as
*x*
gets very large or very
small,
*y*
comes close to 0. Thus,
*f* (*x*) =
has a horizontal
asymptote at
*y* = 0
.

The graph of a function may have several vertical asymptotes.
*f* (*x*) =
has vertical asymptotes of
*x* = 2
and
*x* = - 3
, and
*f* (*x*) =
has vertical asymptotes of
*x* = - 4
and
*x* =
. In general, a vertical asymptote occurs in a rational
function at any value of
*x*
for which the denominator is equal to 0, but for
which the numerator is not equal to 0.