Asymptotes in Rational Functions

A rational function is defined as the quotient of two polynomials:

where Q(x)≠0, These functions often exhibit asymptotes, which are the lines that the graph approaches but never touches. These asymptotes are classified based on how the function behaves near specific values of the input.

Vertical asymptotes occur where the denominator is zero, and the numerator is not, causing the function to be undefined. These are found by solving Q(x)=0. For example:

has a vertical asymptote at x=3, where the graph diverges to infinity.

Horizontal asymptotes describe the function’s behavior as the input becomes very large in magnitude. Their position depends on the degrees of the numerator and denominator:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=3.
• If both degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.
• If the degree of the numerator is greater than that of the denominator, there is no horizontal asymptote.

When the degree of the numerator is exactly one more than the degree of the denominator, a slant asymptote exists. It is found using polynomial division. For instance:

has the slant asymptote y = x - 1.

Rational functions can cross horizontal or slant asymptotes but never vertical ones.