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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:

Horizontal Asymptotes

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:

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.