How many horizontal asymptotes can the graph of y=f(x) have?

1 Answer
Amory W.
Sep 21, 2014

The answer is 0, 1, or 2.

You have to check the end behavior at +-oo, because they don't have to match.

If the growth rate of the numerator is faster than that of the denominator, you won't have a horizontal asymptote. For example, f(x)=x^2, it is implied that the denominator is 1.

If the growth rate of the denominator is faster than that of the numerator, then the horizontal asymptote is y=0. For example, f(x)=1/x.

If the growth rate of the numerator and denominator differ by a constant, c, then the horizontal asymptote is y=c. Here is a graphical example with 2 horizontal asymptotes y=-1 and y=1:
enter image source here

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