Finding Vertical Asymptotes

Pre-Calculus · Axiom Academy

EXAMPLE Finding Vertical Asymptotes Factor, cancel common factors, and set the leftover denominator to zero to locate the vertical asymptotes of a rational function. Find every vertical asymptote of the rational function . Watch out: a factor shared by the top and bottom does not give an asymptote. The curve shoots off to ±∞ only at x = 3 — its single vertical asymptote. At x = 2 the graph has a hole (the canceled factor), not an asymptote. Nice work. You located the vertical asymptotes of a rational function by factoring, canceling, and zeroing the leftover denominator — and you told an asymptote apart from a hole. Factor first: rewrite top and bottom as products before judging where the function blows up. Cancel → hole: a factor shared by numerator and denominator (here x - 2 ) cancels and leaves a hole , never an asymptote. Leftover denominator → asymptotes: set what remains on the bottom to zero. Here x - 3 = 0 , so the only vertical asymptote is x = 3 . Every rational function works the same way: cancel the shared factors, then the surviving denominator's zeros are exactly the vertical asymptotes.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.