Vertical Asymptotes of Rational Functions (F 510)

A rational function is a fraction with polynomials on top and bottom. As x gets closer to a number that makes the denominator zero, you're dividing by something shrinking toward nothing, so the output grows without bound. That's a vertical asymptote: a vertical line x = k that the graph hugs but never touches, because plugging in x = k itself is forbidden — division by zero isn't a number.

Before declaring an asymptote, factor both top and bottom and cancel anything that matches. If a factor like (x - a) appears in both numerator and denominator, it cancels algebraically, and the function is just undefined at x = a rather than blowing up there — that's a hole, not an asymptote. Take f(x) = (x - 1)(x - 3) / ((x - 1)(x + 2)). Cancel (x - 1): you get a hole at x = 1 and a true vertical asymptote at x = -2, since x + 2 never cancels. For f(x) = (x - 1)/(x² - 4), factor the bottom as (x - 2)(x + 2); neither matches the numerator, so vertical asymptotes sit at x = 2 and x = -2. If a factor is squared, like (x - 2)², the graph shoots to positive infinity on both sides instead of flipping sign.

Finding vertical asymptotes and finding domain are the same hunt. Domain excludes every x that zeroes the original, uncancelled denominator — those excluded points are either holes or asymptotes. A graph can cross a horizontal asymptote far out, because that's just a long-run trend, but it can never cross a vertical one, since the function simply doesn't exist there.