Horizontal asymptotes are the limits as x approaches infinity.

If

where c is a constant, then the horizontal asymptote is x = c. Same applies for negative infinity.

You're answers are correct. Whenever you have a polynomial division, where a*x^n and b*x^n are the highest terms for each the numerator and the denominator, than a horizontal asymptote exists at a/b. Note that both n's have to be the same. In other words, this does not apply to 5x^3 / 2x^2.

Edited to add:

if you have a*x^n and b*x^m as the highest terms in the polynomial division, then:

if n > m, the function goes to infinity

if n < m, the function has a horizontal asymptote at x = 0

if n = m, the function has a horizontal asymptote at a/b