AP Calculus AB and BC glossary

Slant asymptote

Also called: Oblique asymptote

A slant asymptote is a non-horizontal line that a curve approaches as x goes to positive or negative infinity. A rational function has one exactly when the degree of the numerator is one more than the degree of the denominator, and you find it by dividing.

y = mx + b \text{ where } \lim_{x \to \infty}\left[f(x) - (mx + b)\right] = 0

Polynomial long division splits the function into a linear part and a remainder that dies off. For $f(x) = \frac{x^{2}+1}{x}$ the division gives $x + \frac{1}{x}$ , and since $\frac{1}{x} \to 0$ , the line $y = x$ is the slant asymptote.

The mistake

Reporting both a horizontal and a slant asymptote. A rational function can have one or the other, never both, because the degree comparison that produces one rules out the other.

Appears in: Unit 1: Limits and Continuity