For a rational function f(x)/g(x), which statement correctly describes its horizontal asymptote based on the degrees of the polynomials?

For a rational function f(x)/g(x), which statement correctly describes its horizontal asymptote based on the degrees of the polynomials?

• A. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
• B. If the degree of the numerator is greater than the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients.
• C. If the degrees are equal, there is no horizontal asymptote.
• D. If the degree of the numerator is less than the degree of the denominator by more than one, the horizontal asymptote is undefined.

Answer: (A)

The end behavior of a rational function is controlled by the degrees of the polynomials. When the numerator’s degree is less than the denominator’s degree, the dominant term in the denominator outgrows the numerator, so as x approaches ±∞ the function tends to 0. That makes the horizontal asymptote y = 0. This describes why the statement is true: the ratio behaves like a small fraction whose value shrinks to zero at the ends. If the degrees were equal, the function would approach the ratio of the leading coefficients, giving a horizontal line y = (leading coefficient of numerator)/(leading coefficient of denominator). If the numerator’s degree is greater, the function grows without bound and there is no horizontal asymptote. The difference in degrees by more than one doesn’t change the fact that the asymptote is still y = 0 when the numerator’s degree is less; the key is the comparison deg(numerator) < deg(denominator).

The end behavior of a rational function is controlled by the degrees of the polynomials. When the numerator’s degree is less than the denominator’s degree, the dominant term in the denominator outgrows the numerator, so as x approaches ±∞ the function tends to 0. That makes the horizontal asymptote y = 0. This describes why the statement is true: the ratio behaves like a small fraction whose value shrinks to zero at the ends. If the degrees were equal, the function would approach the ratio of the leading coefficients, giving a horizontal line y = (leading coefficient of numerator)/(leading coefficient of denominator). If the numerator’s degree is greater, the function grows without bound and there is no horizontal asymptote. The difference in degrees by more than one doesn’t change the fact that the asymptote is still y = 0 when the numerator’s degree is less; the key is the comparison deg(numerator) < deg(denominator).