Шексіздікке ұмтылғанда қисық сызығының асимптотасы

Limit of the tangent line at a point that tends to infinity

In analytic geometry, an asymptote ('/æ//s//ɪ//m//p//t//oʊ//t/) of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity. The word asymptote is derived from the Greek ἀσύμπτωτος (asumptōtos) which means "not falling together", from ἀ priv. + σύν "together" + πτωτ ός "fallen". 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. For curves given by the graph of a function 1=y = ƒ(x), 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. An oblique asymptote has a slope that is non zero but finite, such that the graph of the function approaches it as x tends to +∞ or −∞. 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 the term asymptote by itself is usually reserved for linear asymptotes. 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 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. But these are physical representations of the corresponding mathematical entities; the line and the curve are idealized concepts whose width is 0 (see Line). Therefore, the understanding of the idea of an asymptote requires an effort of reason rather than experience. Consider the graph of the function shown in this section. The coordinates of the points on the curve are of the form where x is a number other than 0. For example, the graph contains the points (1, 1), (2, 0.5), (5, 0.2), (10, 0.1), As the values of become larger and larger, say 100, 1,000, 10,000 , putting them far to the right of the illustration, the corresponding values of , .01, .001, .0001, , become infinitesimal relative to the scale shown. But no matter how large becomes, its reciprocal is never 0, so the curve never actually touches the x axis. Similarly, as the values of become smaller and smaller, say .01, .001, .0001, , making them infinitesimal relative to the scale shown, the corresponding values of , 100, 1,000, 10,000 , become larger and larger. So the curve extends farther and farther upward as it comes closer and closer to the y axis. Thus, both the x and y axis are asymptotes of the curve. These ideas are part of the basis of concept of a limit in mathematics, and this connection is explained more fully below.

The asymptotes most commonly encountered in the study of calculus are of curves of the form 1=y = ƒ(x). These can be computed using limits and classified into horizontal, vertical and oblique asymptotes depending on their orientation. Horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to +∞ or −∞. As the name indicates they are parallel to the x axis. Vertical asymptotes are vertical lines (perpendicular to the x axis) near which the function grows without bound. Oblique asymptotes are diagonal lines such that the difference between the curve and the line approaches 0 as x tends to +∞ or −∞.

The asymptotes of many elementary functions can be found without the explicit use of limits (although the derivations of such methods typically use limits).

When the numerator of a rational function has degree exactly one greater than the denominator, the function has an oblique (slant) asymptote. The asymptote is the polynomial term after dividing the numerator and denominator. This phenomenon occurs because when dividing the fraction, there will be a linear term, and a remainder. For example, consider the function

shown to the right. As the value of x increases, f approaches the asymptote y = x. This is because the other term, 1/(x+1), approaches 0. If the degree of the numerator is more than 1 larger than the degree of the denominator, and the denominator does not divide the numerator, there will be a nonzero remainder that goes to zero as x increases, but the quotient will not be linear, and the function does not have an oblique asymptote.

Asymptotes are used in procedures of curve sketching. An asymptote serves as a guide line to show the behavior of the curve towards infinity. In order to get better approximations of the curve, curvilinear asymptotes have also been used although the term asymptotic curve seems to be preferred.