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