Holes in Rational Graphs

A rational function is a ratio of two polynomials, and it's undefined anywhere the denominator is zero. Usually that gives a vertical asymptote, where the graph shoots off to infinity. But sometimes the numerator is zero at that same x-value too, for the same reason: both polynomials share a factor like (x - a). When that factor cancels completely, the bad behavior cancels with it. What's left is an ordinary-looking graph with one point surgically removed. You'd never spot it by eye because the curve looks perfectly continuous on both sides; only plugging in x = a reveals the function has no value there.

The mechanism is factoring. Write both polynomials in factored form, find the common factor, and cancel it. The simplified expression tells you what the graph looks like almost everywhere, but it's not the same function as the original: the original has a forbidden input that the simplified version doesn't show. You have to carry that restriction forward by hand. The y-coordinate of the hole comes from plugging x = a into the simplified expression, not the original.

Multiplicity decides whether canceling a shared factor leaves a hole or an asymptote. If (x - a) appears to the exact same power top and bottom, it cancels completely and you get a hole. If the denominator has more copies of (x - a) left over after canceling, those leftover factors still blow up the denominator, so you get a vertical asymptote instead. This is why factoring both pieces fully, before doing anything else, is the first move whenever a rational function's behavior at a specific point is in question.