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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