In Exercises 37–44, find the horizontal asymptote, if there is one, of the graph of each rational function. h(x)=12x³/(3x²+1)

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1. Identify the degrees of the numerator and denominator polynomials. For the function $h\left(x\right)=\frac{12x^3}{3x^2+1}$, the degree of the numerator is 3 and the degree of the denominator is 2.
2. Recall the rule for horizontal asymptotes of rational functions: If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (there may be an oblique/slant asymptote instead).
3. Since the degree of the numerator (3) is greater than the degree of the denominator (2), conclude that the function $h\left(x\right)$ does not have a horizontal asymptote.
4. Optionally, to find the behavior of the function as $x\to \; \pm \infty$, consider dividing the numerator by the denominator to find the oblique asymptote, but this is beyond the scope of finding horizontal asymptotes.
5. Summarize: For this function, no horizontal asymptote exists because the numerator's degree is higher than the denominator's.

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, involves analyzing the degrees and leading coefficients of the numerator and denominator polynomials.

Horizontal Asymptotes

Horizontal asymptotes describe the end behavior of a function as x approaches infinity or negative infinity. For rational functions, the horizontal asymptote depends on the degrees of the numerator and denominator polynomials and indicates the value the function approaches but does not necessarily reach.

Degree Comparison of Polynomials

To find horizontal asymptotes, compare the degrees of the numerator and denominator. If the numerator's degree is less, the asymptote is y=0; if equal, it's the ratio of leading coefficients; if greater, there is no horizontal asymptote but possibly an oblique asymptote.