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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.〔("Asymptotes" by Louis A. Talman )〕 In some contexts, such as algebraic geometry, an asymptote is defined as a line which is tangent to a curve 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".〔''Oxford English Dictionary'', second edition, 1989.〕 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.〔D.E. Smith, ''History of Mathematics, vol 2'' Dover (1958) p. 318〕 There are potentially three kinds of asymptotes: ''horizontal'', ''vertical'' and ''oblique'' asymptotes. For curves given by the graph of a function , horizontal asymptotes are horizontal lines that the graph of the function approaches as ''x'' tends to 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 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.〔, §4.18.〕 The study of asymptotes of functions, construed in a broad sense, forms a part of the subject of asymptotic analysis. ==Introduction== 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 to the right. 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, 1000, 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, 1000, 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''-axes 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.〔Reference for section: ("Asymptote" ) ''The Penny Cyclopædia'' vol. 2, The Society for the Diffusion of Useful Knowledge (1841) Charles Knight and Co., London p. 541〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Asymptote」の詳細全文を読む スポンサード リンク
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