Asymptote

In analytic geometry, an asymptote of a curve is a straight 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" derives from the Greek ἀσύμπτωτος, which means "not falling together", from ἀ priv. "not" + 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, 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. 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
More generally, one curve is a [|curvilinear asymptote] of another 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.

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. 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,,,,... 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 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 in mathematics, and this connection is explained more fully below.

Asymptotes of functions

The asymptotes most commonly encountered in the study of calculus are of curves of the form. 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 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 −∞.

Vertical asymptotes

The line x = a is a vertical asymptote of the graph of the function if at least one of the following statements is true:

where is the limit as x approaches the value a from the left, and is the limit as x approaches a from the right.
For example, if ƒ = x/, the numerator approaches 1 and the denominator approaches 0 as x approaches 1. So
and the curve has a vertical asymptote at x = 1.
The function ƒ may or may not be defined at a, and its precise value at the point x = a does not affect the asymptote. For example, for the function
has a limit of +∞ as, ƒ has the vertical asymptote, even though ƒ = 5. The graph of this function does intersect the vertical asymptote once, at. It is impossible for the graph of a function to intersect a vertical asymptote in more than one point. Moreover, if a function is continuous at each point where it is defined, it is impossible that its graph does intersect any vertical asymptote.
A common example of a vertical asymptote is the case of a rational function at a point x such that the denominator is zero and the numerator is non-zero.
If a function has a vertical asymptote, then it isn't necessarily true that the derivative of the function has a vertical asymptote at the same place. An example is
This function has a vertical asymptote at because
and
The derivative of is the function
For the sequence of points
that approaches both from the left and from the right, the values are constantly. Therefore, both one-sided limits of at can be neither nor. Hence doesn't have a vertical asymptote at.

Horizontal asymptotes

Horizontal asymptotes are horizontal lines that the graph of the function approaches as. The horizontal line y = c is a horizontal asymptote of the function y = ƒ if
In the first case, ƒ has y = c as asymptote when x tends to, and in the second ƒ has y = c as an asymptote as x tends to.
For example, the arctangent function satisfies
So the line is a horizontal asymptote for the arctangent when x tends to, and is a horizontal asymptote for the arctangent when x tends to.
Functions may lack horizontal asymptotes on either or both sides, or may have one horizontal asymptote that is the same in both directions. For example, the function has a horizontal asymptote at y = 0 when x tends both to and because, respectively,
Other common functions that have one or two horizontal asymptotes include , the Gaussian function the error function, and the logistic function.

Oblique asymptotes

When a linear asymptote is not parallel to the x- or y-axis, it is called an oblique asymptote or slant asymptote. A function ƒ is asymptotic to the straight line if
In the first case the line is an oblique asymptote of ƒ when x tends to +∞, and in the second case the line is an oblique asymptote of ƒ when x tends to −∞.
An example is ƒ = x + 1/x, which has the oblique asymptote y = x as seen in the limits

Elementary methods for identifying asymptotes

The asymptotes of many elementary functions can be found without the explicit use of limits.

General computation of oblique asymptotes for functions

The oblique asymptote, for the function f, will be given by the equation y = mx + n. The value for m is computed first and is given by
where a is either or depending on the case being studied. It is good practice to treat the two cases separately. If this limit doesn't exist then there is no oblique asymptote in that direction.
Having m then the value for n can be computed by
where a should be the same value used before. If this limit fails to exist then there is no oblique asymptote in that direction, even should the limit defining m exist. Otherwise is the oblique asymptote of ƒ as x tends to a.
For example, the function has
so that is the asymptote of ƒ when x tends to +∞.
The function has
So does not have an asymptote when x tends to +∞.

Asymptotes for rational functions

A rational function has at most one horizontal asymptote or oblique asymptote, and possibly many vertical asymptotes.
The degree of the numerator and degree of the denominator determine whether or not there are any horizontal or oblique asymptotes. The cases are tabulated below, where: is the degree of the numerator, and is the degree of the denominator.

	Asymptotes in general	Example	Asymptote for example
< 0
= 0	y = the ratio of leading coefficients
= 1	y = the quotient of the Euclidean division of the numerator by the denominator
> 1	none		no linear asymptote, but a curvilinear asymptote exists

The vertical asymptotes occur only when the denominator is zero. For example, the following function has vertical asymptotes at x = 0, and x = 1, but not at x = 2.

Oblique asymptotes of rational functions

When the numerator of a rational function has degree exactly one greater than the denominator, the function has an oblique 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/, 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.