Other then looking at the graph.
How do you determine when you have a horizontal, vertical or slant asymptotes.
Use what you learned, back in algebra, about asymptotes:
vertical asymptotes: zeroes of the denominator
horizontal asymptotes: the x-axis, if the degree of the numerator is less than the degree of the denomintor; "y = a/b" for leading coefficients "a" and "b", if the degrees are the same.
slant asymptotes: the quotient (ignoring the remainder) of the long division, assuming the numerator's degree is exactly one greater than the denominator's.