Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4. ƒ(x)=(x^2+1)/(x^2+9)

Key Concepts

Asymptotes

Asymptotes are lines that a graph approaches but never touches. They can be vertical, horizontal, or oblique (slant). Vertical asymptotes occur where a function approaches infinity, typically at values that make the denominator zero. Horizontal asymptotes indicate the behavior of a function as x approaches infinity, while oblique asymptotes occur when the degree of the numerator is one higher than that of the denominator.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. They are typically written in the form f(x) = P(x)/Q(x), where P and Q are polynomials. The behavior of rational functions, especially their asymptotic behavior, is influenced by the degrees of the polynomials in the numerator and denominator. Understanding the degrees helps in determining the presence and type of asymptotes.

Finding Asymptotes

To find asymptotes of a rational function, analyze the degrees of the numerator and denominator. For vertical asymptotes, set the denominator equal to zero and solve for x. For horizontal asymptotes, compare the degrees: if the degree of the numerator is less than the denominator, the asymptote is y=0; if they are equal, the asymptote is y = leading coefficient of P / leading coefficient of Q. If the numerator's degree is greater, check for oblique asymptotes using polynomial long division.