Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4. ƒ(x)=3/(x-5)

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, expressed as f(x) = P(x)/Q(x), where P and Q are polynomials. The behavior of these functions, particularly 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 vertical asymptotes, set the denominator of the rational function equal to zero and solve for x. For horizontal asymptotes, analyze the degrees of the numerator and denominator: if they are equal, the asymptote is y = leading coefficient of P/leading coefficient of Q; if the degree of the numerator is less, y = 0; if greater, there is no horizontal asymptote. Oblique asymptotes can be found using polynomial long division when the numerator's degree is one more than the denominator's.