Holes in Rational Functions

Pre-Calculus · Axiom Academy

LESSON Holes in Rational Functions Why some rational functions have a single point punched out — and how to find exactly where it sits. 1. A Cancelled Factor Punches a Hole A hole (a removable discontinuity ) appears when the same factor sits in the numerator and the denominator. Take . Factor the top and the matching (x-2) cancels — everywhere except at x=2 , where the original fraction is . The simplified form is the line y=x+2 — but with x=2 removed. 2. Where the Hole Sits: the Limit The hole has real coordinates. The x comes from the cancelled factor ( x-2=0 , so x=2 ). The y comes from the simplified form evaluated there: y=x+2 gives 2+2=4 . Slide in toward x=2 from either side and the curve heads straight for (2,4) — that height is the limit, even though the function never actually lands on it. Set the cancelled factor to zero: . Evaluate the simplified form: y=x+2 , so y=4 . The hole is at (2,4) . 3. Hole vs. Vertical Asymptote Both come from the denominator — the difference is whether the factor cancels . Consider (for ). The (x-2) cancels, so x=2 is a hole . The (x-3) survives in the denominator, so x=3 is a vertical asymptote : there the curve runs off to . The hole (cancelled factor): at x=2 the curve is smooth and continuous, just missing the point — an open circle. The asymptote (surviving factor): as the denominator while the numerator doesn't, so — the curve hugs the dashed line x=3 .

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