asymptote

An asymptote is a line that a curve approaches but does not necessarily meet, typically in the limit as the independent variable grows without bound or approaches a point of singularity. More precisely, a line is an asymptote of a curve if the distance between the curve and the line tends to zero under a specified limiting process.

In the context of a function y = f(x), several standard types are recognized. A vertical asymptote occurs at a, if lim_{x→a} f(x) = ±∞, meaning the curve becomes unbounded near x = a and approaches the line x = a. A horizontal asymptote occurs if lim_{x→±∞} f(x) = b, so the curve approaches the line y = b as x grows large in either direction. A slant (or oblique) asymptote is a line y = mx + b with m ≠ 0 for which lim_{x→±∞} [f(x) − (mx + b)] = 0; the curve then approaches the line y = mx + b at infinity.

Examples help illustrate these ideas. The function f(x) = 1/x has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. The rational function f(x) = (2x^2 + 3x + 1)/(x^2 + 1) has a horizontal asymptote y = 2 since the leading terms yield 2 as x → ±∞. The hyperbola y = c/x has vertical asymptotes x = 0 and horizontal asymptote y = 0.

Asymptotic behavior describes how curves behave far away from the origin or near singularities and is fundamental in calculus and analytical studies of functions.