Asymptote Calculator

An asymptote is a line that describes limiting behavior as x approaches a value or grows without bound. A graph may cross a horizontal or slant asymptote, so an asymptote is not defined by never touching the curve. Rational functions can have vertical, horizontal, or polynomial asymptotes depending on their factors and degrees.

Factor P(x) and Q(x), then cancel common factors with matching multiplicity. A denominator factor that remains creates a vertical asymptote at its real root. A factor that cancels completely creates a hole instead. Simply checking whether both polynomials equal zero is not enough when a root has different multiplicities.

Compare the degrees of P and Q. If deg(P) is less than deg(Q), the horizontal asymptote is y = 0. If the degrees are equal, the asymptote is y equal to the ratio of leading coefficients, for example (2x + 5) / (3x - 1) has y = 2 / 3. If deg(P) is greater than deg(Q), there is no horizontal asymptote, though there may be a slant one when the gap is exactly 1.

A slant asymptote is a non-horizontal line y = mx + b that the curve approaches at infinity. It exists only when deg(P) = deg(Q) + 1. To find it, divide P(x) by Q(x) with polynomial long division; the quotient is the slant line and the remainder dies off as x grows. For x^2 / (x - 1), the quotient is x + 1, so the slant asymptote is y = x + 1.

Both start with a denominator root. Cancel common numerator and denominator factors one copy at a time. If the denominator factor cancels completely, the original function has a hole. If any copy remains in the denominator, the root is a vertical asymptote. The calculator compares root multiplicities before classifying each point.