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 ...

, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the $\backslash lim$ symbol (e.g., $\backslash lim\_a\_n$).Courant (1961), p. 29. If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...

ultimately rests.
Limits can be defined in any metric or 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 poin ...

, but are usually first encountered in the 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 ...

s.
History

The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes. Leucippus, Democritus, Antiphon, Eudoxus, and Archimedes developed themethod of exhaustion
The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in are ...

, which uses an infinite sequence of approximations to determine an area or a volume. Archimedes succeeded in summing what is now called a geometric series.
Grégoire de Saint-Vincent gave the first definition of limit (terminus) of a geometric series 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."
Newton dealt with series in his works on ''Analysis with infinite series'' (written in 1669, circulated in manuscript, published in 1711), ''Method of fluxions and infinite series'' (written in 1671, published in English translation in 1736, Latin original published much later) and ''Tractatus de Quadratura Curvarum'' (written in 1693, published in 1704 as an Appendix to his ''Optiks''). In the latter work, Newton considers the binomial expansion of (''x'' + ''o'')Real numbers

In the real numbers, a number $L$ is the limit of the sequence $(x\_n)$, if the numbers in the sequence become closer and closer to $L$, and not to any other number.Examples

*If $x\_n\; =\; c$ for constant ''c'', then $x\_n\; \backslash to\; c$.''Proof'': Choose $N\; =\; 1$. For every $n\; \backslash geq\; N$, $,\; x\_n\; -\; c,\; =\; 0\; <\; \backslash varepsilon$ *If $x\_n\; =\; \backslash frac$, then $x\_n\; \backslash to\; 0$.''Proof'': choose $N\; =\; \backslash left\backslash lfloor\backslash frac\backslash right\backslash rfloor\; +\; 1$ (the floor function). For every $n\; \backslash geq\; N$, $,\; x\_n\; -\; 0,\; \backslash le\; x\_N\; =\; \backslash frac\; <\; \backslash varepsilon$. *If $x\_n\; =\; \backslash frac$ when $n$ is even, and $x\_n\; =\; \backslash frac$ when $n$ is odd, then $x\_n\; \backslash to\; 0$. (The fact that $x\_\; >\; x\_n$ whenever $n$ is odd is irrelevant.) *Given any real number, one may easily construct a sequence that converges to that number by taking decimal approximations. For example, the sequence $0.3,\; 0.33,\; 0.333,\; 0.3333,\; \backslash dots$ converges to $1/3$. Note that the decimal representation $0.3333\backslash dots$ is the ''limit'' of the previous sequence, defined by $$0.3333...\; :\; =\; \backslash lim\_\; \backslash sum\_^n\; \backslash frac$$ *Finding the limit of a sequence is not always obvious. Two examples are $\backslash lim\_\; \backslash left(1\; +\; \backslash tfrac\backslash right)^n$ (the limit of which is the number ''e'') and the Arithmetic–geometric mean. The squeeze theorem is often useful in the establishment of such limits.Definition

We call $x$ the limit of the sequence $(x\_n)$, which is written :$x\_n\; \backslash to\; x$, or :$\backslash lim\_\; x\_n\; =\; x$, if the following condition holds: :For eachreal 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 ...

$\backslash varepsilon\; >\; 0$, 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 every natural number $n\; \backslash geq\; N$, we have $,\; x\_n\; -\; x,\; <\; \backslash varepsilon$.
In other words, for every measure of closeness $\backslash varepsilon$, the sequence's terms are eventually that close to the limit. The sequence $(x\_n)$ is said to converge to or tend to the limit $x$.
Symbolically, this is:
:$\backslash forall\; \backslash varepsilon\; >\; 0\; \backslash left(\backslash exists\; N\; \backslash in\; \backslash N\; \backslash left(\backslash forall\; n\; \backslash in\; \backslash N\; \backslash left(n\; \backslash geq\; N\; \backslash implies\; ,\; x\_n\; -\; x,\; <\; \backslash varepsilon\; \backslash right)\backslash right)\backslash right)$.
If a sequence $(x\_n)$ converges to some limit $x$, then it is convergent and $x$ is the only limit; otherwise $(x\_n)$ is divergent. A sequence that has zero as its limit is sometimes called a null sequence.
Illustration

Properties

Some other important properties of limits of real sequences include the following: *When it exists, the limit of a sequence is unique. *Limits of sequences behave well with respect to the usual arithmetic operations. If $\backslash lim\_\; a\_n$ and $\backslash lim\_\; b\_n$ exists, then ::$\backslash lim\_\; (a\_n\; \backslash pm\; b\_n)\; =\; \backslash lim\_\; a\_n\; \backslash pm\; \backslash lim\_\; b\_n$ ::$\backslash lim\_\; c\; a\_n\; =\; c\; \backslash cdot\; \backslash lim\_\; a\_n$ ::$\backslash lim\_\; (a\_n\; \backslash cdot\; b\_n)\; =\; \backslash left(\backslash lim\_\; a\_n\; \backslash right)\backslash cdot\; \backslash left(\; \backslash lim\_\; b\_n\; \backslash right)$ ::$\backslash lim\_\; \backslash left(\backslash frac\backslash right)\; =\; \backslash frac$ provided $\backslash lim\_\; b\_n\; \backslash ne\; 0$ ::$\backslash lim\_\; a\_n^p\; =\; \backslash left(\; \backslash lim\_\; a\_n\; \backslash right)^p$ *For any continuous function ''f'', if $\backslash lim\_x\_n$ exists, then $\backslash lim\_\; f\; \backslash left(x\_n\; \backslash right)$ exists too. In fact, any real-valued function ''f'' is continuous if and only if it preserves the limits of sequences (though this is not necessarily true when using more general notions of continuity). *If $a\_n\; \backslash leq\; b\_n$ for all $n$ greater than some $N$, then $\backslash lim\_\; a\_n\; \backslash leq\; \backslash lim\_\; b\_n$. *( Squeeze theorem) If $a\_n\; \backslash leq\; c\_n\; \backslash leq\; b\_n$ for all $n$ greater than some $N$, and $\backslash lim\_\; a\_n\; =\; \backslash lim\_\; b\_n\; =\; L$, then $\backslash lim\_\; c\_n\; =\; L$. *( Monotone convergence theorem) If $a\_n$ is bounded and monotonic for all $n$ greater than some $N$, then it is convergent. *A sequence is convergent if and only if every subsequence is convergent. *If every subsequence of a sequence has its own subsequence which converges to the same point, then the original sequence converges to that point. These properties are extensively used to prove limits, without the need to directly use the cumbersome formal definition. For example, once it is proven that $1/n\; \backslash to\; 0$, it becomes easy to show—using the properties above—that $\backslash frac\; \backslash to\; \backslash frac$ (assuming that $b\; \backslash ne\; 0$).Infinite limits

A sequence $(x\_n)$ is said to tend to infinity, written :$x\_n\; \backslash to\; \backslash infty$, or :$\backslash lim\_x\_n\; =\; \backslash infty$, if the following holds: :For every real number $K$, there is a natural number $N$ such that for every natural number $n\; \backslash geq\; N$, we have $x\_n\; >\; K$; that is, the sequence terms are eventually larger than any fixed $K$. Symbolically, this is: :$\backslash forall\; K\; \backslash in\; \backslash mathbb\; \backslash left(\backslash exists\; N\; \backslash in\; \backslash N\; \backslash left(\backslash forall\; n\; \backslash in\; \backslash N\; \backslash left(n\; \backslash geq\; N\; \backslash implies\; x\_n\; >\; K\; \backslash right)\backslash right)\backslash right)$. Similarly, we say a sequence tends to minus infinity, written :$x\_n\; \backslash to\; -\backslash infty$, or :$\backslash lim\_x\_n\; =\; -\backslash infty$, if the following holds: :For every real number $K$, there is a natural number $N$ such that for every natural number $n\; \backslash geq\; N$, we have $x\_n\; <\; K$; that is, the sequence terms are eventually smaller than any fixed $K$. Symbolically, this is: :$\backslash forall\; K\; \backslash in\; \backslash mathbb\; \backslash left(\backslash exists\; N\; \backslash in\; \backslash N\; \backslash left(\backslash forall\; n\; \backslash in\; \backslash N\; \backslash left(n\; \backslash geq\; N\; \backslash implies\; x\_n\; <\; K\; \backslash right)\backslash right)\backslash right)$. If a sequence tends to infinity or minus infinity, then it is divergent. However, a divergent sequence need not tend to plus or minus infinity, and the sequence $x\_n=(-1)^n$ provides one such example.Metric spaces

Definition

A point $x$ of themetric 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 setti ...

$(X,\; d)$ is the limit of the sequence $(x\_n)$ if:
:For each 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 ...

$\backslash varepsilon\; >\; 0$, there is 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 every natural number $n\; \backslash geq\; N$, we have $d(x\_n,\; x)\; <\; \backslash varepsilon$.
Symbolically, this is:
:$\backslash forall\; \backslash varepsilon\; >\; 0\; \backslash left(\backslash exists\; N\; \backslash in\; \backslash N\; \backslash left(\backslash forall\; n\; \backslash in\; \backslash N\; \backslash left(n\; \backslash geq\; N\; \backslash implies\; d(x\_n,\; x)\; <\; \backslash varepsilon\; \backslash right)\backslash right)\backslash right)$.
This coincides with the definition given for real numbers when $X\; =\; \backslash R$ and $d(x,\; y)\; =\; ,\; x-y,$.
Properties

*When it exists, the limit of a sequence is unique, as distinct points are separated by some positive distance, so for $\backslash varepsilon$ less than half this distance, sequence terms cannot be within a distance $\backslash varepsilon$ of both points. *For any continuous function ''f'', if $\backslash lim\_\; x\_n$ exists, then $\backslash lim\_\; f(x\_n)\; =\; f\backslash left(\backslash lim\_x\_n\; \backslash right)$. In fact, a function ''f'' is continuous if and only if it preserves the limits of sequences.Cauchy sequences

A Cauchy sequence is a sequence whose terms ultimately become arbitrarily close together, after sufficiently many initial terms have been discarded. The notion of a Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis. One particularly important result in real analysis is the ''Cauchy criterion for convergence of sequences'': a sequence of real numbers is convergent if and only if it is a Cauchy sequence. This remains true in other complete metric spaces.Topological spaces

Definition

A point $x\; \backslash in\; X$ of the topological space $(X,\; \backslash tau)$ is a or of the sequence $\backslash left(x\_n\backslash right)\_$ if: :For everyneighbourhood
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 a ...

$U$ of $x$, there exists some $N\; \backslash in\; \backslash N$ such that for every $n\; \backslash geq\; N$, we have $x\_n\; \backslash in\; U$.
This coincides with the definition given for metric spaces, if $(X,\; d)$ is a metric space and $\backslash tau$ is the topology generated by $d$.
A limit of a sequence of points $\backslash left(x\_n\backslash right)\_$ in a topological space $T$ is a special case of a limit of a function: the domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...

is $\backslash N$ in the space $\backslash N\; \backslash cup\; \backslash lbrace\; +\; \backslash infty\; \backslash rbrace$, with the induced topology of the affinely extended real number system, the range is $T$, and the function argument $n$ tends to $+\backslash infty$, which in this space is a limit point of $\backslash N$.
Properties

In aHausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ...

, limits of sequences are unique whenever they exist. Note that this need not be the case in non-Hausdorff spaces; in particular, if two points $x$ and $y$ are topologically indistinguishable, then any sequence that converges to $x$ must converge to $y$ and vice versa.
Hyperreal numbers

The definition of the limit using the hyperreal numbers formalizes the intuition that for a "very large" value of the index, the corresponding term is "very close" to the limit. More precisely, a real sequence $(x\_n)$ tends to ''L'' if for every infinite hypernatural ''H'', the term $x\_H$ is infinitely close to ''L'' (i.e., the difference $x\_H\; -\; L$ is infinitesimal). Equivalently, ''L'' is the standard part of $x\_H$: :$L\; =\; (x\_H)$. Thus, the limit can be defined by the formula :$\backslash lim\_\; x\_n=\; (x\_H)$. where the limit exists if and only if the righthand side is independent of the choice of an infinite ''H''.Sequence of more than one index

Sometimes one may also consider a sequence with more than one index, for example, a double sequence $(x\_)$. This sequence has a limit $L$ if it becomes closer and closer to $L$ when both ''n'' and ''m'' becomes very large.Example

*If $x\_\; =\; c$ for constant ''c'', then $x\_\; \backslash to\; c$. *If $x\_\; =\; \backslash frac$, then $x\_\; \backslash to\; 0$. *If $x\_\; =\; \backslash frac$, then the limit does not exist. Depending on the relative "growing speed" of ''n'' and ''m'', this sequence can get closer to any value between 0 and 1.Definition

We call $x$ the double limit of the sequence $(x\_)$, written :$x\_\; \backslash to\; x$, or :$\backslash lim\_\; x\_\; =\; x$, if the following condition holds: :For eachnatural 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 every pair of natural numbers $n,\; m\; \backslash geq\; N$, we have $,\; x\_\; -\; x,\; <\; \backslash varepsilon$.
In other words, for every measure of closeness $\backslash varepsilon$, the sequence's terms are eventually that close to the limit. The sequence $(x\_)$ is said to converge to or tend to the limit $x$.
Symbolically, this is:
:$\backslash forall\; \backslash varepsilon\; >\; 0\; \backslash left(\backslash exists\; N\; \backslash in\; \backslash N\; \backslash left(\backslash forall\; n,\; m\; \backslash in\; \backslash N\; \backslash left(n,\; m\; \backslash geq\; N\; \backslash implies\; ,\; x\_\; -\; x,\; <\; \backslash varepsilon\; \backslash right)\backslash right)\backslash right)$.
Note that the double limit is different from taking limit in ''n'' first, and then in ''m''. The latter is known as iterated limit. Given that both the double limit and the iterated limit exists, they have the same value. However, it is possible that one of them exist but the other does not.
Infinite limits

A sequence $(x\_)$ is said to tend to infinity, written :$x\_\; \backslash to\; \backslash infty$, or :$\backslash lim\_x\_\; =\; \backslash infty$, if the following holds: :For every real number $K$, there is a natural number $N$ such that for every pair of natural numbers $n,m\; \backslash geq\; N$, we have $x\_\; >\; K$; that is, the sequence terms are eventually larger than any fixed $K$. Symbolically, this is: :$\backslash forall\; K\; \backslash in\; \backslash mathbb\; \backslash left(\backslash exists\; N\; \backslash in\; \backslash N\; \backslash left(\backslash forall\; n,\; m\; \backslash in\; \backslash N\; \backslash left(n,\; m\; \backslash geq\; N\; \backslash implies\; x\_\; >\; K\; \backslash right)\backslash right)\backslash right)$. Similarly, a sequence $(x\_)$ tends to minus infinity, written :$x\_\; \backslash to\; -\backslash infty$, or :$\backslash lim\_x\_\; =\; -\backslash infty$, if the following holds: :For every real number $K$, there is a natural number $N$ such that for every pair of natural numbers $n,m\; \backslash geq\; N$, we have $x\_\; <\; K$; that is, the sequence terms are eventually smaller than any fixed $K$. Symbolically, this is: :$\backslash forall\; K\; \backslash in\; \backslash mathbb\; \backslash left(\backslash exists\; N\; \backslash in\; \backslash N\; \backslash left(\backslash forall\; n,\; m\; \backslash in\; \backslash N\; \backslash left(n,\; m\; \backslash geq\; N\; \backslash implies\; x\_\; <\; K\; \backslash right)\backslash right)\backslash right)$. If a sequence tends to infinity or minus infinity, then it is divergent. However, a divergent sequence need not tend to plus or minus infinity, and the sequence $x\_=(-1)^$ provides one such example.Pointwise limits and uniform limits

For a double sequence $(x\_)$, we may take limit in one of the indices, say, $n\; \backslash to\; \backslash infty$, to obtain a single sequence $(y\_m)$. In fact, there are two possible meanings when taking this limit. The first one is called pointwise limit, denoted :$x\_\; \backslash to\; y\_m\backslash quad\; \backslash text$, or :$\backslash lim\_\; x\_\; =\; y\_m\backslash quad\; \backslash text$, which means: :For eachconverges uniformly
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 ...

to $(y\_m)$.
Iterated limit

For a double sequence $(x\_)$, we may take limit in one of the indices, say, $n\; \backslash to\; \backslash infty$, to obtain a single sequence $(y\_m)$, and then take limit in the other index, namely $m\; \backslash to\; \backslash infty$, to get a number $y$. Symbolically, :$\backslash lim\_\; \backslash lim\_\; x\_\; =\; \backslash lim\_\; y\_m\; =\; y$. This limit is known as iterated limit of the double sequence. Note that the order of taking limits may affect the result, i.e., :$\backslash lim\_\; \backslash lim\_\; x\_\; \backslash ne\; \backslash lim\_\; \backslash lim\_\; x\_$ in general. A sufficient condition of equality is given by the Moore-Osgood theorem, which requires the limit $\backslash lim\_x\_\; =\; y\_m$ to be uniform in ''m''.See also

* Limit point * Subsequential limit * Limit superior and limit inferior * Limit of a function * Limit of a sequence of functions * Limit of a sequence of sets * Limit of a net *Pointwise convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared.
Definition
Suppose that X is a set an ...

* Uniform convergence
* Modes of convergence
Notes

Proofs

References

* * * Courant, Richard (1961). "Differential and Integral Calculus Volume I", Blackie & Son, Ltd., Glasgow. * Frank Morley and James Harknessbr>A treatise on the theory of functions(New York: Macmillan, 1893)

External links

*{{Calculus topics Limits (mathematics) Sequences and series