In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a limit is the value that a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
(or
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
) approaches as the input (or index) approaches some
value
Value or values may refer to:
Ethics and social
* Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them
** Values (Western philosophy) expands the notion of value beyo ...
. Limits are essential to
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
and
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, and are used to define
continuity,
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
s, and
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s.
The concept of a
limit of a sequence
As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1."
In mathematics, the limit ...
is further generalized to the concept of a limit of a
topological net, and is closely related to
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
and
direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any categor ...
in
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
.
In formulas, a limit of a function is usually written as
:
(although a few authors may use "Lt" instead of "lim")
and is read as "the limit of of as approaches equals ". The fact that a function approaches the limit as approaches is sometimes denoted by a right arrow (→ or
), as in
:
which reads "
of
tends to
as
tends to
".
History
Grégoire de Saint-Vincent
Grégoire de Saint-Vincent - in latin : Gregorius a Sancto Vincentio, in dutch : Gregorius van St-Vincent - (8 September 1584 Bruges – 5 June 1667 Ghent) was a Flemish Jesuit and mathematician. He is remembered for his work on quadrature of th ...
gave the first definition of limit (terminus) of a
geometric series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series
:\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots
is geometric, because each succ ...
in his work ''Opus Geometricum'' (1647): "The ''terminus'' of a progression is the end of the series, which none progression can reach, even not if she is continued in infinity, but which she can approach nearer than a given segment."
The modern definition of a limit goes back to
Bernard Bolzano
Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Gonzal Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his liber ...
who, in 1817, introduced the basics of the
epsilon-delta
Although the function (sin ''x'')/''x'' is not defined at zero, as ''x'' becomes closer and closer to zero, (sin ''x'')/''x'' becomes arbitrarily close to 1. In other words, the limit of (sin ''x'')/''x'', as ''x'' approaches z ...
technique to define continuous functions. However, his work was not known during his lifetime.
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...
in 1821,
followed by
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
, formalized the definition of the limit of a function which became known as the
(ε, δ)-definition of limit.
The modern notation of placing the arrow below the limit symbol is due to
G. H. Hardy
Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
, who introduced it in his book ''
A Course of Pure Mathematics
''A Course of Pure Mathematics'' is a classic textbook in introductory mathematical analysis, written by G. H. Hardy. It is recommended for people studying calculus. First published in 1908, it went through ten editions (up to 1952) and several r ...
'' in 1908.
Types of limits
In sequences
Real numbers
The expression
0.999... should be interpreted as the limit of the sequence 0.9, 0.99, 0.999, ... and so on. This sequence can be rigorously shown to have the limit 1, and therefore this expression is meaningfully interpreted as having the value 1.
Formally, suppose is a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s. When the limit of the sequence exists, the real number is the ''limit'' of this sequence if and only if for every
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
, there exists a
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
such that for all , we have .
The notation
is often used, and which is read as
:"The limit of ''a
n'' as ''n'' approaches infinity equals ''L''"
The formal definition intuitively means that eventually, all elements of the sequence get arbitrarily close to the limit, since the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
is the distance between and .
Not every sequence has a limit. If it does, then it is called ''
convergent'', and if it does not, then it is ''divergent''. One can show that a convergent sequence has only one limit.
The limit of a sequence and the limit of a function are closely related. On one hand, the limit as approaches infinity of a sequence is simply the limit at infinity of a function —defined on the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
s . On the other hand, if is the domain of a function and if the limit as approaches infinity of is for ''every'' arbitrary sequence of points in which converges to , then the limit of the function as approaches is . One such sequence would be .
Infinity as a limit
There is also a notion of having a limit "at infinity", as opposed to at some finite
. A sequence
is said to "tend to infinity" if, for each real number
, known as the bound, there exists an integer
such that for each
,
That is, for every possible bound, the magnitude of the sequence eventually exceeds the bound. This is often written
or simply
. Such sequences are also called unbounded.
It is possible for a sequence to be divergent, but not tend to infinity. Such sequences are called oscillatory. An example of an oscillatory sequence is
.
For the real numbers, there are corresponding notions of tending to positive infinity and negative infinity, by removing the modulus sign from the above definition:
defines tending to positive infinity, while
defines tending to negative infinity.
Sequences which do not tend to infinity are called bounded. Sequences which do not tend to positive infinity are called bounded above, while those which do not tend to negative infinity are bounded below.
Metric space
The discussion of sequences above is for sequences of real numbers. The notion of limits can be defined for sequences valued in more abstract spaces. One example of a more abstract space is
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s. If
is a metric space with distance function
, and
is a sequence in
, then the limit (when it exists) of the sequence is an element
such that, given
, there exists an
such that for each
, the equation
is satisfied.
An equivalent statement is that
if the sequence of real numbers
.
= Example: ℝn
=
An important example is the space of
-dimensional real vectors, with elements
where each of the
are real, an example of a suitable distance function is the
Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...
, defined by
The sequence of points
converges to
if the limit exists and
.
Topological space
In some sense the ''most'' abstract space in which limits can be defined are
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s. If
is a topological space with topology
, and
is a sequence in
, then the limit (when it exists) of the sequence is a point
such that, given a (open)
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of
, there exists an
such that for every
,
is satisfied.
Function space
This section deals with the idea of limits of sequences of functions, not to be confused with the idea of limits of functions, discussed below.
The field of
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
partly seeks to identify useful notions of convergence on function spaces. For example, consider the space of functions from a generic set
to
. Given a sequence of functions
such that each is a function
, suppose that there exists a function such that for each
,
Then the sequence
is said to
converge pointwise to
. However, such sequences can exhibit unexpected behavior. For example, it is possible to construct a sequence of continuous functions which has a discontinuous pointwise limit.
Another notion of convergence is
uniform convergence
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...
. The uniform distance between two functions
is the maximum difference between the two functions as the argument
is varied. That is,
Then the sequence
is said to uniformly converge or have a uniform limit of
if
with respect to this distance. The uniform limit has "nicer" properties than the pointwise limit. For example, the uniform limit of a sequence of continuous functions is continuous.
Many different notions of convergence can be defined on function spaces. This is sometimes dependent on the
regularity of the space. Prominent examples of function spaces with some notion of convergence are
Lp space
In mathematics, the spaces are function spaces defined using a natural generalization of the Norm (mathematics)#p-norm, -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although ...
s and
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense t ...
.
In functions
Suppose is a
real-valued function
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.
Real-valued functions of a real variable (commonly called ''real fun ...
and is a
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
. Intuitively speaking, the expression
:
means that can be made to be as close to as desired, by making sufficiently close to . In that case, the above equation can be read as "the limit of of , as approaches , is ".
Formally, the definition of the "limit of
as
approaches
" is given as follows. The limit is a real number
so that, given an arbitrary real number
(thought of as the "error"), there is a
such that, for any
satisfying
, it holds that
. This is known as the
(ε, δ)-definition of limit.
The inequality
is used to exclude
from the set of points under consideration, but some authors do not include this in their definition of limits, replacing
with simply
. This replacement is equivalent to additionally requiring that
be continuous at
.
It can be proven that there is an equivalent definition which makes manifest the connection between limits of sequences and limits of functions.
The equivalent definition is given as follows. First observe that for every sequence
in the domain of
, there is an associated sequence
, the image of the sequence under
. The limit is a real number
so that, for ''all'' sequences
, the associated sequence
.
One-sided limit
It is possible to define the notion of having a limit "from above" or "left limit", and a notion of a limit "from below" or "right limit". These need not agree. An example is given by the positive
indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
,
, defined such that
if
, and
if
. At
, the function has a "left limit" of 0, a "right limit" of 1, and its limit does not exist.
Infinity in limits of functions
It is possible to define the notion of "tending to infinity" in the domain of
,
In this expression, the infinity is considered to be signed: either
or
. The "limit of f as x tends to positive infinity" is defined as follows. It is a real number
such that, given any real
, there exists an
so that if
,
. Equivalently, for any sequence
, we have
.
It is also possible to define the notion of "tending to infinity" in the value of
,
The definition is given as follows. Given any real number
, there is a
so that for
, the absolute value of the function
. Equivalently, for any sequence
, the sequence
.
Nonstandard analysis
In
non-standard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta ...
(which involves a
hyperreal
Hyperreal may refer to:
* Hyperreal numbers, an extension of the real numbers in mathematics that are used in non-standard analysis
* Hyperreal.org, a rave culture website based in San Francisco, US
* Hyperreality, a term used in semiotics and po ...
enlargement of the number system), the limit of a sequence
can be expressed as the
standard part
In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every suc ...
of the value
of the natural extension of the sequence at an infinite
hypernatural
In nonstandard analysis, a hyperinteger ''n'' is a hyperreal number that is equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer. An example of an infinite hyperinteger is ...
index ''n=H''. Thus,
:
Here, the standard part function "st" rounds off each finite hyperreal number to the nearest real number (the difference between them is
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
). This formalizes the natural intuition that for "very large" values of the index, the terms in the sequence are "very close" to the limit value of the sequence. Conversely, the standard part of a hyperreal