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
The following graph has a horizontal asymptote of
y = 0:
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:
f (x) =
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 horizontal asymptotes of
x = 2 and
x = - 3, and
f (x) = 
has horizontal 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.
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