The Horizontal Asymptote of a function tells us where the graph will be going when x gets big.
Unlike vertical asymptotes, you can touch or cross a horizontal asymptote.

If the degree of the numerator equals the degree of the denominator, then the horizontal asymptote is

(1)

If the degree of the numerator is less than the degree of the denominator, then it is "proper" and y=0. If the degree of the numerator is greater than the degree of the denominator, then it is improper and you must use long division to write the function as the sum of a polynomial f(x) plus a proper rational function.
Step 1: Put the equation in the y= form
Step 2: Multiply out (expand) any factored polynomials in the numerator or denominator
Step 3: Remove everything except the terms with the biggest exponents of x.

Example:

(2)
\begin{align} f(x)=\frac{4x^2-5x}{x^2-2x}\ \end{align}

Step 1: Both the numerator and denominator are second degree polynomials. Since they are the same degree, we must divide the coefficients of the highest terms.

(3)
\begin{align} \frac{4}{1}=4\ \end{align}