Question

4/(x^2-1) ... does it have a horizontal asymptote ?

Answer 1

Yes. With rational functions, if the degree of the denominator is greater than or equal to the degree of the numerator, then the function has a horizontal asymptote. If the degree of the denominator is greeter than the degree of the numerator, then the horizontal asymptote is the x-axis (y=0); if the degree of the numerator and the degree of the denominator is the same, then the horizontal asymptote is y= the ratio of the leading coefficients.

Answer 2

yes, at y=0

Answer 3

mandolino has it exactly.
if the degree of the numerator is larger there is no horizontal asymptote, although if it is larger by 1, there will be a "slant" asymptote