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 line which is tangent to a curve at infinity.[2][3]

The word asymptote comes from the Greek asymptotos which means "not falling together," from ά priv. + σύν "together" + πτωτ-όσ "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.

A simple example

The idea that a curve may come arbitrarily close to a line without actually becoming the same may seem counter to 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 together, 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 the idea of an asymptote requires an effort of reason rather than experience.

Consider the graph of the equation y=1/x shown to the right. The coordinates of the points on the curve are of the form (x, 1/x) 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 x become larger and larger, say 100, 1000, 10,000 …, putting them far to the right of the illustration, the corresponding values of y, .01, .001, .0001, …, become microscopic relative to the scale shown. But no matter how large x becomes (short of infinity), it's reciprocal 1/x is never 0, so the curve never actually touches the x-axis. Similarly, as the values of x become smaller and smaller, say .01, .001, .0001, …, making them microscopic relative to the scale shown, the corresponding values of y, 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.[7]

Asymptotes of functions

The asymptotes most commonly encountered in the study of calculus are of curves of the form y = ƒ(x). These can be computed using limits and classified into horizontal, vertical and oblique asymptotes depending on its orientation with respect to the X-axis. Horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to +∞ or −∞. As the name indicate 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 so that the difference between the curve and the line approaches 0 as x tends to +∞ or −∞. More general type of asymptotes can be defined in this case.

Vertical asymptotes

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

1. 
2. 

The function ƒ(x) 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 x → 0⁺, ƒ(x) has the vertical asymptote x = 0, even though ƒ(0) = 5. The graph of this function does intersect the vertical asymptote once, at (0,5). It is impossible for the graph of a function to intersect a vertical asymptote (or a vertical line in general) in more than one point.

Horizontal asymptotes

Horizontal asymptotes are horizontal lines that the graph of the function approaches as x → ±∞. The horizontal line y = c is a horizontal asymptote of the function y = ƒ(x) if

or

In the first case, ƒ(x) has y = c as asymptote when x tends to −∞, and in the second that ƒ(x) has y = c as an asymptote as x tends to +∞

For example the arctangent function satisfies

and

So the line y = −π/2 is a horizontal tangent for the arctangent when x tends to −∞, and y = π/2 is a horizontal tangent 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 ƒ(x) = 1/(x²+1) has a horizontal asymptote at y = 0 when x tends both to −∞ and +∞ because, respectively,

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 f(x) is asymptotic to the straight line y = mx + n (m ≠ 0) if

In the first case the line y = mx + n is an oblique asymptote of ƒ(x) when x tends to −∞, and in the second case the line y = mx + n is an oblique asymptote of ƒ(x) when x tends to −∞

An example is ƒ(x) = x−1/x, which has the oblique asymptote y = x (m = 1, c = 0) as seen in the limits

Elementary methods for identifying asymptotes

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

General computation of oblique asymptotes for functions

The oblique asymptote, for the function f(x), 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 limit defining m exist. Otherwise y = mx + n is the oblique asymptote of ƒ(x) as x tends to a.

For example, the function ƒ(x) = (2x² + 3x + 1)/x has

and then

so that y = 2x + 3 is the asymptote of ƒ(x) when x tends to +∞. The function ƒ(x) = ln x has

and then , which does not exist.

So y = ln x does not have an asymptote when x tends to +∞.

Asymptotes for rational functions

A rational function has at most one horizontal asymptote or oblique (slant) 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 deg(numerator) is the degree of the numerator, and deg(denominator) is the degree of the denominator.

Table listing the cases of horizontal and oblique asymptotes for rational functions deg(numerator) − deg(denominator) asymptotes example asymptote
< 0 y = 0
= 0 y = the ratio of leading coefficients
= 1 y = the quotient, in the long division of the fraction
> 1 none

The vertical asymptotes occur only when the denominator is zero (If both the numerator and denominator are zero, the multiplicities of the zero are compared). 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 (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, y = 1/(x+1) becomes smaller.

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.

Transformations of known functions

If a known function has an asymptote (such as y=0 for f(x)=e^x), then the translations of it also have an asymptote.

• If x=a is a vertical asymptote of f(x), then x=a+h is a vertical asymptote of f(x-h)+k
• If y=c is a horizontal asymptote of f(x), then y=c+k is a horizontal asymptote of f(x-h)+k

If a known function has an asymptote, then the scaling of the function also have an asymptote.