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.