In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the symbol (e.g., ).^{[1]} If such a limit exists, the sequence is called convergent.^{[2]} A sequence that does not converge is said to be divergent.^{[3]} The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.^{[1]}
n | n sin(1/n) |
---|---|
1 | 0.841471 |
2 | 0.958851 |
... | |
10 | 0.998334 |
... | |
100 | 0.999983 |
Limits can be defined in any metric or topological space, but are usually first encountered in the real numbers.
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.
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."^{[4]}
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 the limit as o tends to 0.
In the 18th century, mathematicians such as 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 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 in the 1870s.
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.
We call the limit of the sequence if the following condition holds:
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
Symbolically, this is:
If a sequence converges to some limit then it is convergent and is the only limit; otherwise is divergent. A sequence that has zero as its limit is sometimes called a null sequence.
Example of a sequence which converges to the limit
Regardless which we have, there is an index so that the sequence lies afterwards completely in the epsilon tube
There is also for a smaller an index so that the sequence is afterwards inside the epsilon tube
For each there are only finitely many sequence members outside the epsilon tube.
Limits of sequences behave well with respect to the usual arithmetic operations. If and then and, if neither b nor any is zero, ^{[5]}
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 (provided, in each equation below, that the limits on the right exist).
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 it becomes easy to show—using the properties above—that (assuming that ).
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 minus infinity, then it is divergent. However, a divergent sequence need not tend to plus or minus infinity, and the sequence provides one such example.
A point of the metric space is the limit of the sequence if for all there is an 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.
A point of the topological space is a limit or limit point^{[7]}^{[8]} of the sequence if for every neighbourhood of there exists some such that for every ^{[9]} This coincides with the definition given for metric spaces, if is a metric space and is the topology generated by
A limit of a sequence of points in a topological space is a special case of a limit of a function: the domain is in the space with the induced topology of the affinely extended real number system, the range is and the function argument tends to which in this space is a limit point of
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 and are topologically indistinguishable, then any sequence that converges to must converge to and vice versa.
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.
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 is infinitely close to L (i.e., the difference is infinitesimal). Equivalently, L is the standard part of
Thus, the limit can be defined by the formula