...have more than two horizontal asymptotes?
I just want to know for sure.
You find horizontal asymptotes by considering the limiting behavior of y = f(x) as x approaches +oo and -oo:
If then and the line is a horizontal asymptote.
If then and the line is a horizontal asymptote.
When they exist, the values of a and b are unique by the 'uniquesness of a limit theorem'.
Therefore there are no more than two possible horizontal asymptotes.
Stick at it