Limit of a sequence

Limit of a sequence
n n sin(1/n)
1 0.841471
2 0.958851
10 0.998334
100 0.999983

As the positive integer n becomes larger and larger, the value n sin(1/n) becomes arbitrarily close to 1. We say that "the limit of the sequence n sin(1/n) equals 1."

The limit of a sequence (xn) is, intuitively, the unique number or point L (if it exists) such that the terms of the sequence become arbitrarily close to L for "large" values of n. If the limit exists, then we say that the sequence is convergent and that it converges to L.

Convergence of sequences is a fundamental notion in mathematical analysis, which has been studied since ancient times.


Formal definition

  • For a sequence of real numbers \left( x_n:n\in \mathbb{N}\right)\;
A real number L is said to be the limit of the sequence xn, written
\lim_{n \to \infty} x_n=L,
if and only if for every real number ε > 0, there exists a natural number N such that for every n > N we have | xnL | < ε.

An element LT is said to be a limit of this sequence if and only if for every neighborhood S of L there is a natural number N such that xnS for all n > N. In this generality a sequence may admit more than one limit, but if T is a Hausdorff space, then each sequence has at most one limit. When a unique limit L exists, it may be written
\lim_{n \to \infty} x_n = L.

If a sequence has a limit, we say the sequence is convergent, and that the sequence converges to the limit. Otherwise, the sequence is divergent (see also oscillation).

A null sequence is a sequence that converges to 0.

Hyperreal definition

A sequence xn tends to L if for every infinite hypernatural H, the term xH is infinitely close to L, i.e., the difference xH - L is infinitesimal. Equivalently, L is the standard part of xH

 L = {\rm st}(x_H)\,.

Thus, the limit can be defined by the formula

\lim_{n \to \infty} x_n= {\rm st}(x_H),

where the limit exists if and only if the righthand side is independent of the choice of an infinite H.


The definition means that eventually all elements of the sequence get as close as we want to the limit. (The condition that the elements become arbitrarily close to all of the following elements does not, in general, imply the sequence has a limit. See Cauchy sequence).

A sequence of real numbers may tend to +\infty or -\infty, compare infinite limits. Even though this can be written in the form

 \lim_{n \to \infty} x_n = \infty and  \lim_{n \to \infty} x_n = -\infty

such a sequence is called divergent, unless we explicitly consider it a sequence in the affinely extended real number system or (in the first case only) the real projective line. In the latter cases the sequence has a limit (in the space itself), so could be called convergent, but when using this term here, care should be taken that this does not cause confusion.

The limit of a sequence of points \left(x_n:n\in \mathbb{N}\right)\; in a topological space T is a special case of the limit of a function: the domain is \mathbb{N} in the space \mathbb{N} \cup \lbrace +\infty \rbrace with the induced topology of the affinely extended real number system, the range is T, and the function argument n tends to +∞, which in this space is a limit point of \mathbb{N}.


  • The sequence 1, -1, 1, -1, 1, ... is oscillatory.
  • The series with partial sums 1/2, 1/2 + 1/4, 1/2 + 1/4 + 1/8, 1/2 + 1/4 + 1/8 + 1/16, ... converges with limit 1. This is an example of an infinite series.
  • If a is a real number with absolute value | a | < 1, then the sequence an has limit 0. If 0 < a, then the sequence a1/n has limit 1.


\lim_{n\to\infty} \frac{1}{n^p} = 0 \hbox{ if } p > 0
\lim_{n\to\infty} a^n = 0 \hbox{ if } |a| < 1
\lim_{n\to\infty} n^{\frac{1}{n}} = 1
\lim_{n\to\infty} a^{\frac{1}{n}} = 1 \hbox{ if } a>0


Consider the following function: f( x ) = xn if n-1 < xn. Then the limit of the sequence of xn is just the limit of f( x) at infinity.

A function f, defined on a first-countable space, is continuous if and only if it is compatible with limits in that (f(xn)) converges to f(L) given that (xn) converges to L, i.e.

\lim_{n\to\infty}x_n=L implies \lim_{n\to\infty}f(x_n)=f(L)

Note that this equivalence does not hold in general for spaces which are not first-countable.

Compare the basic property (or definition):

f is continuous at x if and only if \lim_{x\to L}f(x)=f(L)

A subsequence of the sequence (xn) is a sequence of the form (xa(n)) where the a(n) are natural numbers with a(n) < a(n+1) for all n. Intuitively, a subsequence omits some elements of the original sequence. A sequence is convergent if and only if all of its subsequences converge towards the same limit.

Every convergent sequence in a metric space is a Cauchy sequence and hence bounded. A bounded monotonic sequence of real numbers is necessarily convergent: this is sometimes called the fundamental theorem of analysis. More generally, every Cauchy sequence of real numbers has a limit, or short: the real numbers are complete.

A sequence of real numbers is convergent if and only if its limit superior and limit inferior coincide and are both finite.

The algebraic operations are continuous everywhere (except for division around zero divisor); thus, given

\lim_{n \to \infty}x_n = L_1
\lim_{n \to \infty}y_n = L_2


\lim_{n \to \infty}(x_n+y_n) = L_1 + L_2

\lim_{n \to \infty}(x_ny_n) = L_1L_2

and (if L2 and yn are non-zero)

\lim_{n \to \infty}(x_n/y_n) = L_1/L_2

These rules are also valid for infinite limits using the rules

  • q + ∞ = ∞ for q ≠ -∞
  • q × ∞ = ∞ if q > 0
  • q × ∞ = -∞ if q < 0
  • q / ∞ = 0 if q ≠ ± ∞

(see extended real number line).


The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes.

Leucippus, Democritus, Antiphon, Eudoxus and Archimedes developed the method of exhaustion, 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.

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)n which he then linearizes by taking limits (letting o→0).

In the 18th century, mathematicians like Euler 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 in his Théorie des fonctions analytiques (1797) opined that the lack of rigour precluded further development in calculus. Gauss in his etude of hypergeometric series (1813) for the first time rigorously investigated under which conditions a series converged to a limit.

The modern definition of a limit (for any ε there exists an index N so that ...) was given independently by Bernhard Bolzano (Der binomische Lehrsatz, Prague 1816, little noticed at the time) and by Cauchy in his Cours d'analyse (1821).

See also


External links

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Limit superior and limit inferior — In mathematics, the limit inferior (also called infimum limit, liminf, inferior limit, lower limit, or inner limit) and limit superior (also called supremum limit, limsup, superior limit, upper limit, or outer limit) of a sequence can be thought… …   Wikipedia

  • Limit (mathematics) — This is an overview of the idea of a limit in mathematics. For specific uses of a limit, see limit of a sequence and limit of a function. In mathematics, the concept of a limit is used to describe the value that a function or sequence approaches… …   Wikipedia

  • Limit of a function — x 1 0.841471 0.1 0.998334 0.01 0.999983 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. It is said that the limit of (sin x)/x as x approache …   Wikipedia

  • Sequence — For other uses, see Sequence (disambiguation). In mathematics, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length …   Wikipedia

  • Séquence principale — Le diagramme de Hertzsprung Russell figure les étoiles. En abscisse, l indice de couleur (B V) ; en ordonnée, la magnitude absolue. La séquence principale se voit comme une bande diagonale marquée allant du haut à gauche au bas à droite. Ce… …   Wikipédia en Français

  • limit — 1. verb /ˈlɪmɪt/ a) To restrict; not to allow to go beyond a certain bound. We need to limit the power of the executive. b) To have a limit in a particular set. Im limiting myself to two drinks tonight. Syn: cap, restrict …   Wiktionary

  • Limit — A limit can be: * Limit (mathematics), including: ** Limit of a function ** Limit of a sequence ** One sided limit ** Limit superior and limit inferior ** Limit of a net ** Limit point ** Limit (category theory) * A constraint (mathematical,… …   Wikipedia

  • Limit point — In mathematics, a limit point (or accumulation point) of a set S in a topological space X is a point x in X that can be approximated by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point …   Wikipedia

  • Limit set — In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they can …   Wikipedia

  • limit — I. noun Etymology: Middle English, from Anglo French limite, from Latin limit , limes boundary Date: 14th century 1. a. something that bounds, restrains, or confines b. the utmost extent 2. a. a geographic or political boundary b. plural the… …   New Collegiate Dictionary

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”