Editing Asymptote (section)

==Introduction==
[[File:Hyperbola one over x.svg|right|thumb|300px|<math>f(x)=\tfrac{1}{x}</math> graphed on [[Cartesian coordinates]]. The ''x'' and ''y''-axis 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. But these are physical representations of the corresponding mathematical entities; the line and the curve are idealized concepts whose width is 0 (see [[Line (geometry)|Line]]). Therefore, the understanding of the idea of an asymptote requires an effort of reason rather than experience.

Consider the graph of the function <math>f(x) = \frac{1}{x}</math> shown in this section. The coordinates of the points on the curve are of the form <math>\left(x, \frac{1}{x}\right)</math> where x is a number other than 0. For example, the graph contains the points (1,&thinsp;1), (2, 0.5), (5, 0.2), (10, 0.1), ... As the values of <math>x</math> become larger and larger, say 100, 1,000, 10,000 ..., putting them far to the right of the illustration, the corresponding values of <math>y</math>, .01, .001, .0001, ..., become infinitesimal relative to the scale shown. But no matter how large <math>x</math> becomes, its reciprocal <math>\frac{1}{x}</math> is never 0, so the curve never actually touches the ''x''-axis. Similarly, as the values of <math>x</math> become smaller and smaller, say .01, .001, .0001, ..., making them infinitesimal relative to the scale shown, the corresponding values of <math>y</math>, 100, 1,000, 10,000 ..., become larger and larger. So the curve extends further and further 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 of a function|limit]] in mathematics, and this connection is explained more fully below.<ref>Reference for section: [https://books.google.com/books?id=HTwi2M37rQAC&pg=PA541 "Asymptote"] ''[[Penny Cyclopædia|The Penny Cyclopædia]]'' vol. 2, The Society for the Diffusion of Useful Knowledge (1841) Charles Knight and Co., London p. 541</ref>