...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 $\displaystyle \lim_{x \rightarrow -\infty} f(x) = a$ then $\displaystyle y \rightarrow a$ and the line $\displaystyle y = a$ is a horizontal asymptote.
If $\displaystyle \lim_{x \rightarrow +\infty} f(x) = b$ then $\displaystyle y \rightarrow b$ and the line $\displaystyle y = b$ 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