For a rational function where the degree of the numerator equals the degree of the denominator, the horizontal asymptote is given by:

For a rational function where the degree of the numerator equals the degree of the denominator, the horizontal asymptote is given by:

• A. The ratio of the leading coefficients
• B. y = 0
• C. No horizontal asymptote
• D. The slope of the oblique asymptote

Answer: (A)

When the degrees of the numerator and denominator are the same, the dominant terms in the rational function determine its end behavior. As x grows large in magnitude, the lower-degree terms become insignificant, so the function behaves like the ratio of the leading terms. That means the horizontal asymptote is the constant you get by dividing the leading coefficient of the numerator by the leading coefficient of the denominator. For instance, f(x) = (3x + 2)/(2x − 5) approaches 3/2 as x → ±∞, so y = 3/2 is the horizontal asymptote. The other scenarios don’t apply here: y = 0 would require the numerator to have lower degree than the denominator; no horizontal asymptote would occur if the numerator’s degree were higher; and an oblique (sloped) asymptote happens when the numerator’s degree is exactly one more than the denominator’s, not when they’re equal.

When the degrees of the numerator and denominator are the same, the dominant terms in the rational function determine its end behavior. As x grows large in magnitude, the lower-degree terms become insignificant, so the function behaves like the ratio of the leading terms. That means the horizontal asymptote is the constant you get by dividing the leading coefficient of the numerator by the leading coefficient of the denominator. For instance, f(x) = (3x + 2)/(2x − 5) approaches 3/2 as x → ±∞, so y = 3/2 is the horizontal asymptote. The other scenarios don’t apply here: y = 0 would require the numerator to have lower degree than the denominator; no horizontal asymptote would occur if the numerator’s degree were higher; and an oblique (sloped) asymptote happens when the numerator’s degree is exactly one more than the denominator’s, not when they’re equal.