AP Calculus AB and BC glossary

Rational function

A rational function is a ratio of two polynomials, continuous wherever the denominator is nonzero. Each zero of the denominator gives a hole if that factor cancels completely out of the denominator, and a vertical asymptote if it appears more times in the denominator than in the numerator.

Write $f(x) = \frac{p(x)}{q(x)}$ with $p$ and $q$ polynomials and $q$ not the zero polynomial. Since polynomials are continuous everywhere, the only places a rational function can misbehave are the zeros of $q$ , and factoring both parts sorts every one of them into a hole or a vertical asymptote.

f(x) = \frac{x^{2} - 9}{x^{2} - 3x} = \frac{(x-3)(x+3)}{x(x-3)}

The denominator is zero at $x = 3$ and $x = 0$ . At $x = 3$ the factor cancels and the limit is 2, so the graph has a hole at the point $(3, 2)$ . At $x = 0$ nothing cancels, so that is a vertical asymptote. Multiplicity settles the close calls: a factor appearing more times in the denominator than in the numerator survives cancellation and still gives an asymptote.

The mistake

Reporting every zero of the denominator as a vertical asymptote. In the function above, $x = 3$ is a hole, a removable discontinuity where the limit exists and is finite. The mirror error is cancelling first and then reading the domain off $\frac{x+3}{x}$ , which looks perfectly healthy at $x = 3$ even though the original function is undefined there.

Appears in: Unit 2: Defining the Derivative