Holes vs Asymptotes

Algebra 2 · Axiom Academy

Two vertical breaks that look nothing alike — and the single factoring step that tells them apart. 1. A Hole: When a Factor Cancels A hole — a removable discontinuity — appears when the numerator and denominator share a common factor. At the zero of that shared factor the function reads : undefined, but only at that one point. Both top and bottom carry the factor (x-2) Cancel it — but x=2 stays forbidden 2. A Vertical Asymptote: When a Factor Survives If a zero of the denominator does not cancel with anything on top, you have a nonzero number divided by something shrinking to zero — so the size explodes. The graph shoots off to or and never crosses the line. Nothing on top to cancel the (x-3) As , the denominator is a tiny negative number, so . As , the denominator is a tiny positive number, so . 3. The Test: Factor, Cancel, Read the Leftovers Every hole-or-asymptote question is settled the same way. Factor the top and bottom completely, cancel what they share, and look at what each zero became. Factor the numerator and denominator completely. Hole: the zero of each cancelled factor. Asymptote: each zero left in the denominator. From the cancelled factor (x+1) . Point: — the value of the simplified at x=-1 . From the surviving denominator factor (x-2) . 4. Side by Side: Same Pole, Opposite Behavior Put both cases at the very same value, x=2 . On the left the factor cancels — ; on the right it doesn't — . That single difference is everything.

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