In
analytic geometryAnalytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...
, an
asymptote (icon) of a
curveIn mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
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. In some contexts, such as
algebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, an asymptote is defined as a line which is
tangentIn geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...
to a curve at infinity.
The word asymptote is derived from the
GreekGreek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
asymptotos which means "not falling together," from ἀ
priv.In linguistics, abessive , caritive and privative are names for a grammatical case expressing the lack or absence of the marked noun...
+ σύν "together" + πτωτ-ός "fallen." The term was introduced by
Apollonius of PergaApollonius of Perga [Pergaeus] was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes...
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 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 usually the term
asymptote by itself is 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 analysisIn mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The methodology has applications across science. Examples are...
.
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
LineThe notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
). Therefore the understanding of 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 infinitesimal relative to the scale shown. But no matter how large
x becomes, its 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 infinitesimal 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
limitIn mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....
in mathematics, and this connection is explained more fully below.
Asymptotes of functions
The asymptotes most commonly encountered in the study of
calculusCalculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
are of curves of the form . These can be computed using
limitsIn mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
and classified into
horizontal,
vertical and
oblique asymptotes depending on its orientation. 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 if at least one of the following statements is true:
-
-
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 ,
ƒ(
x) has the vertical asymptote , 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 . 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 is a horizontal tangent for the arctangent when
x tends to −∞, and 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 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 (
m ≠ 0) if
In the first case the line is an oblique asymptote of
ƒ(
x) when
x tends to +∞, and in the second case the line 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,
n = 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 the limit defining
m exist. Otherwise is the oblique asymptote of
ƒ(
x) as
x tends to
a.
For example, the function has
and then
so that is the asymptote of
ƒ(
x) when
x tends to +∞. The function has
and then
, which does not exist.
So does not have an asymptote when
x tends to +∞.
Asymptotes for rational functions
A
rational functionIn mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
has at most one horizontal asymptote or oblique (slant) asymptote, and possibly many vertical asymptotes.
The
degreeThe degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...
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.
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
dividingIn algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised version of the familiar arithmetic technique called long division...
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)=
ex), 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)
- If y=c is a horizontal asymptote of f(x), then y=c+k is a horizontal asymptote of f(x)+k
If a known function has an asymptote, then the
scaling of the function also have an asymptote.
- If y=ax+b is an asymptote of f(x), then y=cax+cb is an asymptote of cf(x)
For example,
f(
x)=
ex-1+2 has horizontal asymptote
y=0+2=2, and no vertical or oblique asymptotes.
General definition
Let be a parametric plane curve, in coordinates
A(
t) = (
x(
t),
y(
t)). Suppose that the curve tends to infinity, that is:
A line ℓ is an asymptote of
A if the distance from the point
A(
t) to ℓ tends to zero as
t →
b.
For example, the upper right branch of the curve
y = 1/
x can be defined parametrically as
x =
t,
y = 1/
t (where
t>0). First,
x → ∞ as
t → ∞ and the distance from the curve to the
x-axis is 1/
t which approaches 0 as
t → ∞. Therefore the
x-axis is an asymptote of the curve. Also,
y → ∞ as
t → 0 from the right, and the distance between the curve and the
y-axis is
t which approaches 0 as
t → 0. So the
y-axis is also an asymptote. A similar argument shows that the lower left branch of the curve also has the same two lines as asymptotes.
Although the definition here uses a parameterization of the curve, the notion of asymptote does not depend on the parameterization. In fact, if the equation of the line is
then the distance from the point
A(
t) = (
x(
t),
y(
t)) to the line is given by
if γ(
t) is a change of parameterization then the distance becomes
which tends to zero simultaneously as the previous expression.
An important case is when the curve is the
graphIn mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...
of a real function (a function of one real variable and returning real values). The graph of the function
y =
ƒ(
x) is the set of points of the plane with coordinates (
x,
ƒ(
x)). For this, a parameterization is
This parameterization is to be considered over the open intervals (
a,
b), where
a can be −∞ and
b can be +∞.
An asymptote can be either vertical or non-vertical (oblique or horizontal). In the first case its equation is
x =
c, for some real number
c. The non-vertical case has equation , where
m and
are real numbers. All three types of asymptotes can be present at the same time in specific examples. Unlike asymptotes for curves that are graphs of functions, a general curve may have more than two non-vertical asymptotes, and may cross its vertical asymptotes more than once.
Curvilinear asymptotes
Let be a parametric plane curve, in coordinates
A(
t) = (
x(
t),
y(
t)), and
B be another (unparameterized) curve. Suppose, as before, that the curve
A tends to infinity. The curve
B is a curvilinear asymptote of
A if the shortest of the distance from the point
A(
t) to a point on
B tends to zero as
t →
b. Sometimes
B is simply referred to as an asymptote of
A, when there is no risk of confusion with linear asymptotes.
For example, the function
has a curvilinear asymptote , which is known as a
parabolic asymptote because it is a
parabolaIn mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
rather than a straight line.
Asymptotes and curve sketching
The notion of asymptote is related to procedures of
curve sketchingIn geometry, curve sketching includes techniques that can used to produce a rough idea of overall shape of a plane curve given its equation without computing the large numbers of points required for a detailed plot...
. An asymptote serves as a guide line that serves to show the behavior of the curve towards infinity. In order to get better approximations of the curve, asymptotes that are general curves have also been used although the term asymptotic curve seems to be preferred.
Algebraic curves
The asymptotes of an
algebraic curveIn algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
in the
affine planeIn mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
are the lines that are tangent to the projectivized curve through a point at infinity. Asymptotes are often considered only for real curves, although they also make sense when defined in this way for curves over an arbitrary
fieldIn abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
.
A plane curve of degree
n intersects its asymptote at most at
n−2 other points, by
Bézout's theoremBézout's theorem is a statement in algebraic geometry concerning the number of common points, or intersection points, of two plane algebraic curves. The theorem claims that the number of common points of two such curves X and Y is equal to the product of their degrees...
, as the intersection at infinity is of multiplicity at least two. For a conic, there are a pair of lines that do not intersect the conic at any complex point: these are the two asymptotes of the conic.
A plane algebraic curve is defined by an equation of the form
P(
x,
y) = 0 where
P is a polynomial of degree
n
where
Pk is
homogeneousIn mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial...
of degree
k. Vanishing of the linear factors of the highest degree term
Pn defines the asymptotes of the curve: if , then the line
is an asymptote, where
t0 is chosen so that the curve and line meet at infinity. Over the complex numbers,
Pn splits into linear factors, each of which defines an asymptote. However, over the reals, not only may
Pn fail to split, but also if a linear factor has multiplicity greater than one, the resulting asymptote may be entirely spurious. For example, the curve has no real points in the finite plane, but its highest order term gives the asymptote
x = 0 with multiplicity 4.
Other uses of the term
The hyperbolas
have asymptotes
The equation for the union of these two lines is
Similarly, the
hyperboloids
are said to have the
asymptotic cone
The distance between the hyperboloid and cone approaches 0 as the distance from the origin approaches infinity.
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