Limit of a sequence

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diagram of a hexagon and pentagon circumscribed outside a circle
The sequence given by the perimeters of regular n-sided polygons that circumscribe the unit circle has a limit equal to the perimeter of the circle, i.e. . The corresponding sequence for inscribed polygons has the same limit.
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."

In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to".[1] If such a limit exists, the sequence is called convergent. A sequence which does not converge is said to be divergent.[2] The limit of a sequence is said to be the fundamental notion on which the whole of analysis ultimately rests.[1]

Limits can be defined in any metric or topological space, but are usually first encountered in the real numbers.







Real numbers


In the real numbers, a number is the limit of the sequence if the numbers in the sequence become closer and closer to and not to any other number.



Formal Definition


In other words, for every measure of closeness , the sequence's terms are eventually that close to the limit. The sequence is said to converge to or tend to the limit , written or .

If a sequence converges to some limit, then it is convergent; otherwise it is divergent.


Limits of sequences behave well with respect to the usual arithmetic operations. If and , then , and, if neither b nor any is zero, .

For any continuous function f, if then . 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).

Some other important properties of limits of real sequences include the following.

These properties are extensively used to prove limits without the need to directly use the cumbersome formal definition. Once proven that it becomes easy to show that , (), using the properties above.

Infinite limits

A sequence is said to tend to infinity, written or if, for every K, there is an N such that, for every , ; that is, the sequence terms are eventually larger than any fixed K. Similarly, if, for every K, there is an N such that, for every , . If a sequence tends to infinity, or to minus infinity, then it is divergent (however, a divergent sequence need not tend to plus or minus infinity).

Metric spaces


A point x of the metric space (X, d) is the limit of the sequence (xn) if, for all ε > 0, there is an N such that, for every , . This coincides with the definition given for real numbers when and .


For any continuous function f, if then . In fact, a 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 less than half this distance, sequence terms cannot be within a distance of both points.

Topological spaces


A point x of the topological space (X, τ) is the limit of the sequence (xn) if, for every neighbourhood U of x, there is an N such that, for every , . This coincides with the definition given for metric spaces if (X,d) is a metric space and is the topology generated by d.

The limit of a sequence of points in a topological space T is a special case of the limit of a function: the domain is in the space 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 .


If X is a Hausdorff space then limits of sequences are unique where they exist. Note that this need not be the case in general; in particular, if two points x and y are topologically indistinguishable, any sequence that converges to x must converge to y and vice-versa.

Cauchy sequences




A sequence is convergent if and only if it is Cauchy.

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 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


Thus, the limit can be defined by the formula

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

See also


  1. 1.0 1.1 Courant (1961), p. 29.
  2. Courant (1961), p. 39.



External links

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