Rational functions
A zero of the numerator gives an x-intercept, and a zero of the denominator gives a vertical asymptote.
A worked example
f(x) = (x - 6)/(x + 8) has a vertical asymptote at x = -8, an x-intercept at x = 6, and a horizontal asymptote at y = 1 since the degrees match.
The mistake people actually make
Set the denominator equal to zero and solve. For x + 7 the asymptote is at x = -7, not x = 7. When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.