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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.

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