Family Overview

In Plain Language

A hole (removable discontinuity) is a missing point on an otherwise well-behaved rational graph. It happens at an input $x=c$ that makes both numerator and denominator zero, when the multiplicity of $c$ in the numerator is greater than or equal to its multiplicity in the denominator.

Then nearby outputs approach a single finite height $L$, so the hole is at the point $(c,L)$. In limit language: $\lim_{x\to c}r(x)=L$ (and the left and right limits match).

If instead the denominator multiplicity is greater, you get a vertical asymptote (Topic 1.9), not a hole.

Why It Matters

AP Precalculus asks students to decide hole vs VA by comparing multiplicities, then find the hole’s coordinates by simplifying and evaluating (or by examining nearby outputs). Mixing holes with asymptotes is a high-frequency error.

How Families Can Help

• Ask: “Is this factor on top and bottom? Which side has more copies?”
• Practice: cancel what you can, then plug $c$ into the simplified expression for $L$.
• Contrast: hole = finite limit; VA = infinite blow-up.

What Not to Do

• Do not call every canceled factor’s $x$-value a zero of $r$ (still undefined at the hole).
• Do not treat a hole as a vertical asymptote.
• Do not forget to find the $y$-coordinate $L$.
• Do not assume left and right limits differ at a hole (for standard rational holes they agree).