How to find vertical asymptotes and horizontal asymptotes of a rational function?

A vertical asymptote occurs when the graph of a function approaches infinity as the independent variable approaches one or more specific rational values. This happens when the denominator of a fraction becomes zero. However, the fraction has to be in its simplest form (no common monomial factor between denominator and numerator).
A horizontal asymptote usually occurs for large positive or negative values of the independent variable where the function is a fraction and the polynomials forming the numerator and denominator have the same degree.
Example: y=(x+7)/(x-1) has a vertical asymptote at x=1 and a horizontal asymptote at y=1.
Example: y=(3x²+5x+9)/(x²-1) has a vertical asymptote at x=1 and at x=-1, and a horizontal asymptote at y=3.