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A vertical asymptote occurs at *x* = *c* when the following are all true

1) f (c) is undefined | |||

2) f (x) = ∞ or - ∞ | |||

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 | |||

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.

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