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
:
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
:
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.
When a value of
x
sets both the denominator and the numerator of a
rational function equal to 0, there is a hole in the graph; that is, a
single point at which the function has no value.
f (x) =
has a hole at
x = 2
:
Examples: Name the vertical asymptotes and holes in the graphs of the
following equations: