Find the horizontal asymptote, if there is one, of the graph of each rational function. g(x)=12x²/(3x²+1)

Verified step by step guidance

1. Identify the degrees of the numerator and denominator polynomials in the rational function \(g(x) = \frac{12x^2}{3x^2 + 1}\). The degree of the numerator is 2, and the degree of the denominator is also 2.
2. Recall the rule for horizontal asymptotes of rational functions: If the degrees of numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients.
3. Determine the leading coefficient of the numerator, which is 12 (from \$12x^2$), and the leading coefficient of the denominator, which is 3 (from \$3x^2$).
4. Write the horizontal asymptote as \(y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}} = \frac{12}{3}\).
5. Simplify the fraction to express the horizontal asymptote in simplest form.

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding the behavior of rational functions, especially their graphs, is essential for analyzing asymptotes and limits.

Horizontal Asymptotes

A horizontal asymptote describes the behavior of a function as x approaches infinity or negative infinity. It is a horizontal line y = L that the graph approaches but does not necessarily touch, indicating the end behavior of the function.

Degree of Polynomials in Rational Functions

The degrees of the numerator and denominator polynomials determine the horizontal asymptote. If degrees are equal, the asymptote is the ratio of leading coefficients; if numerator degree is less, asymptote is y=0; if greater, no horizontal asymptote exists.