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

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Identify the degrees of the numerator and the denominator. The degree of the numerator is 3, and the degree of the denominator is 2.

Compare the degrees of the numerator and the denominator. Since the degree of the numerator (3) is greater than the degree of the denominator (2), there is no horizontal asymptote.

For rational functions where the degree of the numerator is greater than the degree of the denominator, the graph does not have a horizontal asymptote.

Instead, the function may have an oblique (slant) asymptote, which can be found using polynomial long division.

Perform polynomial long division of the numerator by the denominator to find the equation of the oblique asymptote, if needed.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the given function h(x) = 12x^3 / (3x^2 + 1), the numerator is a polynomial of degree 3, and the denominator is a polynomial of degree 2. Understanding the degrees of the polynomials is crucial for analyzing the behavior of the function 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 greater than the degree of the denominator, the horizontal asymptote is at y = 0; if they are equal, the asymptote is at the ratio of the leading coefficients.

Leading Coefficients

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. In the function h(x), the leading coefficient of the numerator (12) and the leading coefficient of the denominator (3) are essential for determining the horizontal asymptote when the degrees of the numerator and denominator are equal. This concept helps in simplifying the function's behavior at extreme values of x.

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