Limit (mathematics)
Euclid's Elements, written around 300 BCE, contains Proposition X.1 which describes a process of subtracting more than half from a larger magnitude repeatedly. This ancient technique, known as the Method of exhaustion, allowed mathematicians to approximate areas and volumes by getting closer and closer to a true value without ever quite reaching it in finite steps. Archimedes later used this method to calculate the area of a circle with remarkable precision. The core idea was that if you keep removing pieces larger than half of what remains, eventually the leftover piece becomes smaller than any given magnitude. Grégoire de Saint-Vincent provided the first formal definition of a limit for geometric series in his 1647 work Opus Geometricum. He described the terminus as an end point that no progression can reach even if continued into infinity, yet one that can be approached nearer than any specific segment. Isaac Newton offered a clear definition of limits in the Scholium to his 1687 Principia Mathematica. He stated that ultimate ratios are not actually ratios of ultimate quantities but limits which they can approach so closely that their difference is less than any given quantity. Bruce Pourciau argues that Newton possessed a sophisticated understanding of these concepts and may have provided the first epsilon argument.
Bernard Bolzano developed the basics of the epsilon-delta technique in 1817 to define continuous functions. His work remained unknown to other mathematicians until thirty years after his death. Augustin-Louis Cauchy followed up on this foundation in 1821, and Karl Weierstrass later formalized the definition of the limit of a function. This became known as the epsilon-delta definition of limit. The modern notation placing the arrow below the limit symbol was invented by John Gaston Leathem in 1905. G. H. Hardy popularized this notation in his 1908 textbook A Course of Pure Mathematics. Formally, for every real number epsilon greater than zero, there exists a delta such that if x is within delta of c, then f(x) is within epsilon of L. This inequality excludes c from the set of points under consideration in some definitions. Some authors replace the strict inequality with simply x not equal to c. This replacement is equivalent to additionally requiring that f be continuous at c. The expression means that the value of the function can be made arbitrarily close to L by choosing x sufficiently close to c. The common notation reads as the limit of f of x as x approaches c equals L.
The discussion of sequences above applies to sequences of real numbers. The notion of limits can be defined for sequences valued in more abstract spaces like metric spaces. If X is a metric space with distance function d and s is a sequence in X, the limit when it exists is an element l such that given epsilon there exists an N such that for each n greater than N we have d(sn, l) less than epsilon. An equivalent statement is that sn converges to l if the sequence of real numbers d(sn, l) converges to 0. Consider the space of n-dimensional real vectors where elements are tuples of real numbers. A suitable distance function is the Euclidean distance defined by the square root of the sum of squared differences. The sequence of points converges to l if the limit exists and the distance between them goes to zero. There is also a notion of having a limit tend to infinity rather than to a finite value. A sequence sn is said to tend to infinity if for each real number M known as the bound, there exists an integer N such that for each n greater than N, sn is greater than M. That is, for every possible bound, the sequence eventually exceeds the bound. It is possible for a sequence to be divergent but not tend to infinity. Such sequences are called oscillatory. An example of an oscillatory sequence is (-1)^n.
In some sense the most abstract space in which limits can be defined are topological spaces. If X is a topological space with topology T and s is a sequence in X, the limit when it exists is a point x such that given an open neighborhood U of x there exists an N such that for every n greater than N, sn is in U. In this case, the limit if it exists may not be unique. However it must be unique if X is a Hausdorff space. Let X be a sequence in a topological space S. For concreteness, S can be thought as R^n but the definitions hold more generally. The limit set is the set of points such that if there is a convergent subsequence snk with limit l then l belongs to the limit set. A use of this notion is to characterize the long-term behavior of oscillatory sequences. Consider the sequence (-1)^n starting from n equals 1. The first few terms are -1, 1, -1, 1. It can be checked that it is oscillatory so has no limit, but has limit points {-1, 1}. This notion is used in dynamical systems to study limits of trajectories. Defining a trajectory to be a function f: R -> M, the point f(t) is thought of as the position of the trajectory at time t. The limit set of a trajectory is defined as follows. To any sequence of increasing times tn there is an associated sequence of positions f(tn). If x is the limit set of the sequence f(tn) for any sequence of increasing times, then x is a limit set of the trajectory.
Limits are used to define a number of important concepts in analysis including continuity derivatives and integrals. A particular expression of interest which is formalized as the limit of a sequence is sums of infinite series. These are infinite sums of real numbers generally written as sum from k equals 0 to infinity of ak. This is defined through limits as follows. Given a sequence of real numbers ak, the sequence of partial sums is defined by sn equal to sum from k equals 0 to n of ak. If the limit of the sequence sn exists, the value of the expression is defined to be the limit. Otherwise, the series is said to be divergent. A classic example is the Basel problem where ak equals 1 over k squared. Then the sum equals pi squared over 6. However while for sequences there is essentially a unique notion of convergence, for series there are different notions of convergence. This is due to the fact that the expression does not discriminate between different orderings of the sequence ak. The convergence properties of the sequence of partial sums can depend on the ordering of the sequence. A series which converges for all orderings is called unconditionally convergent. It can be proven to be equivalent to absolute convergence. A surprising result for conditionally convergent series is the Riemann series theorem. Depending on the ordering, the partial sums can be made to converge to any real number as well as positive or negative infinity.
For sequences of real numbers, a number of properties can be proven. Suppose sn and tn are two sequences converging to s and t respectively. The sum of limits is equal to limit of sum. The product of limits is equal to limit of product. The inverse of limit is equal to limit of inverse as long as t is not zero. Equivalently, the function f(x) equals 1/x is continuous about nonzero x. A property of convergent sequences of real numbers is that they are Cauchy sequences. The definition of a Cauchy sequence sn is that for every real number epsilon greater than zero, there is an N such that whenever m and n are greater than N, d(sn, sm) is less than epsilon. Informally, for any arbitrarily small error epsilon, it is possible to find an interval of diameter epsilon such that eventually the sequence is contained within the interval. In general metric spaces, it continues to hold that convergent sequences are also Cauchy. But the converse is not true: not every Cauchy sequence is convergent in a general metric space. A classic counterexample is the rational numbers Q with the usual distance. The sequence of decimal approximations to square root of 2 truncated at the nth decimal place is a Cauchy sequence but does not converge in Q. A metric space in which every Cauchy sequence is also convergent is known as a complete metric space. One reason Cauchy sequences can be easier to work with than convergent sequences is that they are a property of the sequence alone while convergent sequences require not just the sequence but also the limit of the sequence.
Common questions
Who wrote Euclid's Elements and what method did it contain?
Euclid wrote the Elements around 300 BCE which contains Proposition X.1 describing the Method of exhaustion. This technique allowed mathematicians to approximate areas and volumes by getting closer to a true value without reaching it in finite steps.
When was the first formal definition of a limit for geometric series provided?
Grégoire de Saint-Vincent provided the first formal definition of a limit for geometric series in his 1647 work Opus Geometricum. He described the terminus as an end point that no progression can reach even if continued into infinity yet one that can be approached nearer than any specific segment.
What is the epsilon-delta definition of limit and who developed its basics?
Bernard Bolzano developed the basics of the epsilon-delta technique in 1817 to define continuous functions while Augustin-Louis Cauchy followed up on this foundation in 1821. Karl Weierstrass later formalized the definition of the limit of a function known as the epsilon-delta definition of limit.
How do limits apply to sequences in metric spaces and topological spaces?
In a metric space with distance function d, the limit when it exists is an element l such that given epsilon there exists an N such that for each n greater than N we have d(sn, l) less than epsilon. In a topological space with topology T, the limit when it exists is a point x such that given an open neighborhood U of x there exists an N such that for every n greater than N, sn is in U.
Why are some series conditionally convergent but not unconditionally convergent?
A series which converges for all orderings is called unconditionally convergent and proven equivalent to absolute convergence. The Riemann series theorem states that depending on the ordering, partial sums of conditionally convergent series can be made to converge to any real number as well as positive or negative infinity.