What is the difference between a vertical and a horizontal asymptote

Definition of vertical asymptote

A vertical asymptote is a line x = a that the graph approaches but never crosses as the function values become arbitrarily large in magnitude. It appears when the denominator of a rational function equals zero at x = a and the factor (x‑a) does not cancel with the numerator. For example, f(x)=$\frac{2x}{x-3}$ has denominator zero at x=3, and because (x‑3) is not a factor of the numerator, the line x=3 is a vertical asymptote. Formally, \lim_{x\to3^-}f(x)=-$\infty$ and \lim_{x\to3^+}f(x)=+$\infty$, and the graph never crosses the line x=3 but can approach it arbitrarily close.

Definition of horizontal asymptote

A horizontal asymptote is a line y = b that the graph approaches as x moves toward positive or negative infinity. Whether such a line exists depends on the degrees of the numerator (n) and denominator (m) of the rational function. If n < m, the horizontal asymptote is y = 0. If n = m, the asymptote is y = leading coefficient of numerator divided by leading coefficient of denominator. If n > m, there is no horizontal asymptote, only an oblique one. For instance, g(x)=$\frac{3x^{2}+1}{2x^{2}-5}$ has equal degrees, so its horizontal asymptote is y = $\frac{3}{2}$. Formally, \lim_{x\to\infty}g(x)=$\frac{3}{2}$ and \lim_{x\to-$\infty$}g(x)=$\frac{3}{2}$, and the function may cross its horizontal asymptote at finite x values but settles toward it as |x| grows.

Key differences between vertical and horizontal asymptotes:

• Vertical asymptotes are vertical lines (x = a); horizontal asymptotes are horizontal lines (y = b).
• Vertical asymptotes arise from non‑canceled zeros of the denominator; horizontal asymptotes arise from degree comparison of numerator and denominator.
• Near a vertical asymptote the function magnitude grows without bound; near a horizontal asymptote the function values settle to a constant.

How to find each asymptote:

1. 1Factor numerator and denominator completely.
2. 2Cancel any common factors.
3. 3For vertical asymptotes, set each remaining denominator factor equal to zero.
4. 4For horizontal asymptotes, compare the highest powers of x in numerator and denominator and apply the degree rules.

Summary of horizontal asymptote cases based on degree: