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

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1. Identify the degrees of the numerator and denominator polynomials in the rational function g(x) = $\frac{12x^2}{3x^2+1}$. The numerator has degree 2 and the denominator has degree 2.
2. Recall the rule for horizontal asymptotes of rational functions: If the degrees of numerator and denominator are equal, the horizontal asymptote is y = (leading coefficient of numerator) / (leading coefficient of denominator).
3. Identify the leading coefficients: For the numerator, it is 12; for the denominator, it is 3.
4. Write the horizontal asymptote as y = $\frac{12}{3}$ based on the leading coefficients.
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 Comparison of Polynomials

To find horizontal asymptotes of rational functions, compare the degrees of the numerator and denominator polynomials. 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.