In Exercises 37–44, find the horizontal asymptote, if there is one, of the graph of each rational function. f(x)=12x/(3x^2+1)

Key Concepts

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the given function f(x) = 12x/(3x^2 + 1), the numerator is a polynomial of degree 1, and the denominator is a polynomial of degree 2. Understanding the structure of rational functions is essential for analyzing their behavior, particularly as x approaches infinity.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as the input approaches infinity or negative infinity. For rational functions, the horizontal asymptote can be determined by comparing 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 = 0; if they are equal, it is the ratio of their leading coefficients.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. In the context of rational functions, the degrees of the numerator and denominator are crucial for determining the horizontal asymptote. For the function f(x) = 12x/(3x^2 + 1), the degree of the numerator is 1 and the degree of the denominator is 2, which influences the asymptotic behavior of the function.