mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...

, the limit of a sequence is the value that the terms of a sequence
In , a sequence is an enumerated collection of in which repetitions are allowed and matters. Like a , it contains (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unl ...

"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
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical ...

ultimately rests.
Limits can be defined in any metric
METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...

or topological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

, but are usually first encountered in the real number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s.
History

The Greek philosopher is famous for formulating paradoxes that involve limiting processes.Leucippus
Leucippus (; el, Λεύκιππος, ''Leúkippos''; fl. 5th century BCE) is reported in some ancient sources to have been a philosopher
A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the grc, φιλό ...

, Democritus
Democritus (; el, Δημόκριτος, ''Dēmókritos'', meaning "chosen of the people"; – ) was an Ancient Greek
Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient w ...

, Antiphon
An antiphon (Greek language, Greek ἀντίφωνον, ἀντί "opposite" and φωνή "voice") is a short chant in Christianity, Christian ritual, sung as a refrain. The texts of antiphons are the Psalms. Their form was favored by St Ambrose a ...

, Eudoxus, and Archimedes
Archimedes of Syracuse (; grc, ; ; ) was a Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its popula ...

developed the method of exhaustion
The method of exhaustion (; ) is a method of finding the area
Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface o ...

, 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
In mathematics, a geometric series (mathematics), series is the sum of an infinite number of Summand, terms that have a constant ratio between successive terms. For example, 1/2 + 1/4 + 1/8 + 1/16 + · · ·, the series
:\frac \,+\, \frac \,+\, ...

.
Newton
Newton most commonly refers to:
* Isaac Newton (1642–1726/1727), English scientist
* Newton (unit), SI unit of force named after Isaac Newton
Newton may also refer to:
Arts and entertainment
* Newton (film), ''Newton'' (film), a 2017 Indian fil ...

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'')mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

s such as Euler
Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

succeeded in summing some ''divergent'' series by stopping at the right moment; they did not much care whether a limit existed, as long as it could be calculated. At the end of the century, Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaRoman_Forum.html" ;"title="Curia Julia in the Roman Forum">Curia Julia in the Roman Forum
A senate is a deliberative assembly, often the upper house or Debating chamber, chamber of a bicame ...

in his ''Théorie des fonctions analytiques'' (1797) opined that the lack of rigour precluded further development in calculus. Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician
This is a List of German mathematician
A mathematician is someone who uses an extensive knowledge of m ...

in his etude of hypergeometric series
In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a Special functions, special function represented by the hypergeometric series, that includes many other special functions as special case, speci ...

(1813) for the first time rigorously investigated the conditions under which a series converged to a limit.
The modern definition of a limit (for any ε there exists an index ''N'' so that ...) was given by Bernhard Bolzano (''Der binomische Lehrsatz'', Prague 1816, which was little noticed at the time), and by Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematics, mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university withou ...

in the 1870s.
Real numbers

In thereal numbers
Real may refer to:
* Reality, the state of things as they exist, rather than as they may appear or may be thought to be
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish col ...

, a number $L$ is the limit of the sequence
In , a sequence is an enumerated collection of in which repetitions are allowed and matters. Like a , it contains (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unl ...

$(x\_n),$ 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 $N\; =\; \backslash left\backslash lfloor\backslash frac\backslash right\backslash rfloor\; +\; 1$ (thefloor function
In mathematics and computer science, the floor function is the function (mathematics), function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted \operatorname(x) or \lfloor x\rfloor ...

). For every $n\; \backslash geq\; N,$ $,\; x\_n\; -\; 0,\; \backslash le\; x\_N\; =\; \backslash frac\; <\; \backslash varepsilon.$
*If $x\_n\; =\; 1/n$ 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
A decimal representation of a non-negative
In mathematics, the sign of a real number is its property of being either positive, negative number, negative, or zero. Depending on local conventions, zero may be considered as being neither positive n ...

$0.3333...$ 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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. The squeeze theorem
In calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zer ...

is often useful in the establishment of such limits.
Formal definition

We call $x$ the limit of thesequence
In , a sequence is an enumerated collection of in which repetitions are allowed and matters. Like a , it contains (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unl ...

$(x\_n)$ if the following condition holds:
*For each real number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

$\backslash varepsilon\; >\; 0,$ there exists a natural number
File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...)
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, o ...

$N$ 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 $x,$ written $x\_n\; \backslash to\; x$ or $\backslash lim\_\; x\_n\; =\; x.$
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

Limits of sequences behave well with respect to the usualarithmetic operations
Arithmetic (from the Greek ἀριθμός ''arithmos'', 'number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so for ...

. If $a\_n\; \backslash to\; a$ and $b\_n\; \backslash to\; b,$ then $a\_n+b\_n\; \backslash to\; a+b,$ $a\_n\backslash cdot\; b\_n\; \backslash to\; ab$ and, if neither ''b'' nor any $b\_n$ is zero, $\backslash frac\; \backslash to\; \backslash frac.$
For any continuous function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

''f'', if $x\_n\; \backslash to\; x$ then $f(x\_n)\; \backslash to\; f(x).$ In fact, any real-valued function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

''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).
Some other important properties of limits of real sequences include the following (provided, in each equation below, that the limits on the right exist).
*The limit of a sequence is unique.
*$\backslash lim\_\; (a\_n\; \backslash pm\; b\_n)\; =\; \backslash lim\_\; a\_n\; \backslash pm\; \backslash lim\_\; b\_n$
*$\backslash lim\_\; c\; a\_n\; =\; c\; \backslash cdot\; \backslash lim\_\; a\_n$
*$\backslash lim\_\; (a\_n\; \backslash cdot\; b\_n)\; =\; (\backslash lim\_\; a\_n)\backslash cdot(\; \backslash lim\_\; b\_n)$
*$\backslash lim\_\; \backslash left(\backslash frac\backslash right)\; =\; \backslash frac$ provided $\backslash lim\_\; b\_n\; \backslash ne\; 0$
*$\backslash lim\_\; a\_n^p\; =\; \backslash left;\; href="/html/ALL/s/\backslash lim\_\_a\_n\_\backslash right.html"\; ;"title="\backslash lim\_\; a\_n\; \backslash right">\backslash lim\_\; a\_n\; \backslash right$
*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 $n\; >\; N,$ and $\backslash lim\_\; a\_n\; =\; \backslash lim\_\; b\_n\; =\; L,$ then $\backslash lim\_\; c\_n\; =\; L.$
*If a sequence is Sequence#Bounded, bounded and Sequence#Increasing and decreasing, monotonic, 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 for every ''K'', there is an ''N'' such that for every $n\; \backslash geq\; N,$ $x\_n\; >\; K$; that is, the sequence terms are eventually larger than any fixed ''K''. Similarly, $x\_n\; \backslash to\; -\backslash infty$ if for every ''K'', there is an ''N'' such that for every $n\; \backslash geq\; N,$ $x\_n\; <\; K.$ 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 the metric space $(X,\; d)$ is the limit of theProperties

For anycontinuous function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

''f'', if $x\_n\; \backslash to\; x$ then $f(x\_n)\; \backslash to\; f(x).$ In fact, a Function (mathematics), function ''f'' is continuous if and only if it preserves the limits of sequences.
Limits of sequences are unique when they exist, as distinct points are separated by some positive distance, so for $\backslash epsilon$ less than half this distance, sequence terms cannot be within a distance $\backslash epsilon$ of both points.
Topological spaces

Definition

A point $x\; \backslash in\; X$ of the topological space $(X,\; \backslash tau)$ is a or of theProperties

In a Hausdorff space, 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.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 space, complete metric spaces.Definition in 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 function, standard part of $x\_H$ $$L\; =\; (x\_H).\backslash ,$$ 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''.See also

* * * Limit superior and limit inferior * Modes of convergence * Net (mathematics)#Limits of nets, Limit of a net — A net (mathematics), net is a topological generalization of a sequence. * Set-theoretic limit * Shift rule *Notes

Proofs

References

* * * Richard Courant, Courant, Richard (1961). "Differential and Integral Calculus Volume I", Blackie & Son, Ltd., Glasgow. * Frank Morley and James Harkness (mathematician), James Harkness]A treatise on the theory of functions

(New York: Macmillan, 1893)

External links

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