A asymptotes of a curve pdf intersecting an asymptote infinitely many times. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors.

In some contexts, such as algebraic geometry, an asymptote is defined as a line which is tangent to a curve at infinity. The term was introduced by Apollonius of Perga in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve. There are three kinds of asymptotes: horizontal, vertical and oblique asymptotes. Vertical asymptotes are vertical lines near which the function grows without bound.

Asymptotes convey information about the behavior of curves in the large, and determining the asymptotes of a function is an important step in sketching its graph. The study of asymptotes of functions, construed in a broad sense, forms a part of the subject of asymptotic analysis. The x and y-axes are the asymptotes.

The idea that a curve may come arbitrarily close to a line without actually becoming the same may seem to counter everyday experience. The representations of a line and a curve as marks on a piece of paper or as pixels on a computer screen have a positive width. So if they were to be extended far enough they would seem to merge, at least as far as the eye could discern. Therefore, the understanding of the idea of an asymptote requires an effort of reason rather than experience.