Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4. ƒ(x)=(2x+6)/(x-4)

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 describe linear behavior when the degree of the numerator is one higher than that of the denominator.

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 numerator and denominator. Understanding the degrees helps in determining the presence and type of asymptotes, which are critical for graphing the function accurately.

To find vertical asymptotes, set the denominator equal to zero and solve for x. For horizontal asymptotes, compare the degrees of the numerator and denominator: if the degrees are equal, the asymptote is y = leading coefficient of P / leading coefficient of Q; if the degree of the numerator is less, the asymptote is y = 0. For oblique asymptotes, perform polynomial long division when the numerator's degree is one higher than the denominator's, yielding a linear equation.