In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors.^{[1]} In some contexts, such as algebraic geometry, an asymptote is defined as a line which is tangent to a curve at infinity.^{[2]}^{[3]}
The word asymptote is derived from the Greek asymptotos which means "not falling together," from ἀ priv. + σύν "together" + πτωτός "fallen."^{[4]} 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.^{[5]}
For curves given by the graph of a function y = ƒ(x), there are potentially three kinds of asymptotes: horizontal, vertical and oblique asymptotes. Horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to +∞ or −∞. Vertical asymptotes are vertical lines near which the function grows without bound.
More generally, one curve is a curvilinear asymptote of another (as opposed to a linear asymptote) if the distance between the two curves tends to zero as they tend to infinity, although usually the term asymptote by itself is reserved for linear asymptotes. Asymptotes convey much of the information about the behavior of curves in the large, and determining the asymptotes of a function is an important step in sketching its graph.^{[6]} The study of asymptotes of functions, construed in a broad sense, forms a part of the subject of asymptotic analysis.
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