Limit (mathematics)

In mathematics, a limit is the value that a function (or sequence) approaches as the argument (or index) approaches some value. Limits of functions are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory. The limit inferior and limit superior provide generalizations of the concept of a limit which are particularly relevant when the limit at a point may not exist.

In formulas, a limit of a function is usually written as \[\lim_{x \to c} f(x) = L,\] and is read as "the limit of \(f\) of \(x\) as \(x\) approaches \(c\) equals \(L\)". This means that the value of the function \(f\) can be made arbitrarily close to \(L\), by choosing \(x\) sufficiently close to \(c\). Alternatively, the fact that a function \(f\) approaches the limit \(L\) as \(x\) approaches \(c\) is sometimes denoted by a right arrow (→ or \(\rightarrow\)), as in \[f(x) \to L \text{ as } x \to c,\] or in

\[f(x) \xrightarrow[x \to c]{} L,\]

which reads "\(f\) of \(x\) tends to \(L\) as \(x\) tends to \(c\)".

According to Hankel (1871), the modern concept of limit originates from Proposition X.1 of Euclid's Elements, which forms the basis of the Method of exhaustion found in Euclid and Archimedes: "Two unequal magnitudes being set out, if from the greater there is subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process is repeated continually, then there will be left some magnitude less than the lesser magnitude set out."

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."

In the Scholium to Principia in 1687, Isaac Newton had a clear definition of a limit, stating that "Those 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 further argues that, in addition to Newton actually having a more sophisticated understanding of limits than he is generally credited with, he also provided the first epsilon argument.

The modern definition of a limit goes back to Bernard Bolzano who, in 1817, developed the basics of the epsilon-delta technique to define continuous functions. However, his work remained unknown to other mathematicians until thirty years after his death.

Augustin-Louis Cauchy in 1821, followed by Karl Weierstrass, formalized the definition of the limit of a function which became known as the (ε, δ)-definition of limit.

The modern notation of placing the arrow below the limit symbol was invented by John Gaston Leathem in 1905 and popularized by G. H. Hardy's 1908 textbook A Course of Pure Mathematics.

Suppose f is a real-valued function and c is a real number. Intuitively speaking, the expression

\[\lim_{x \to c}f(x) = L\]

means that f(x) can be made to be as close to L as desired, by making x sufficiently close to c. In that case, the above equation can be read as "the limit of f of x, as x approaches c, is L".

Formally, the definition of the "limit of \(f(x)\) as \(x\) approaches \(c\)" is given as follows. The limit is a real number \(L\) so that, given an arbitrary real number \(\varepsilon > 0\) (thought of as the "error"), there is a \(\delta > 0\) such that, for any \(x\) satisfying \(0 < |x - c| < \delta\), it holds that \(| f(x) - L | < \varepsilon\). This is known as the (ε, δ)-definition of limit.

The inequality \(0 < |x - c|\) is used to exclude \(c\) from the set of points under consideration, but some authors do not include this in their definition of limits, replacing \(0 < |x - c| < \delta\) with simply \(|x - c| < \delta\). This replacement is equivalent to additionally requiring that \(f\) be continuous at \(c\).

It can be proven that there is an equivalent definition which makes manifest the connection between limits of sequences and limits of functions. The equivalent definition is given as follows. First observe that for every sequence \(\{x_n\}\) in the domain of \(f\), there is an associated sequence \(\{f(x_n)\}\), the image of the sequence under \(f\). The limit is a real number \(L\) so that, for all sequences \(x_n \rightarrow c\), the associated sequence \(f(x_n) \rightarrow L\).

In non-standard analysis (which involves a hyperreal enlargement of the number system), the limit of a sequence \((a_n)\) can be expressed as the standard part of the value \(a_H\) of the natural extension of the sequence at an infinite hypernatural index n = H. Thus, \[\lim_{n \to \infty} a_n = \operatorname{st}(a_H) .\] Here, the standard part function "st" rounds off each finite hyperreal number to the nearest real number (the difference between them is infinitesimal). This formalizes the natural intuition that for "very large" values of the index, the terms in the sequence are "very close" to the limit value of the sequence. Conversely, the standard part of a hyperreal \(a=[a_n]\) represented in the ultrapower construction by a Cauchy sequence \((a_n)\), is simply the limit of that sequence: \[\operatorname{st}(a)=\lim_{n \to \infty} a_n .\] In this sense, taking the limit and taking the standard part are equivalent procedures.

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 \[\sum_{n = 1}^\infty a_n.\] This is defined through limits as follows: given a sequence of real numbers \(\{a_n\}\), the sequence of partial sums is defined by \[s_n = \sum_{i = 1}^n a_i.\] If the limit of the sequence \(\{s_n\}\) exists, the value of the expression \(\sum_{n = 1}^\infty a_n\) is defined to be the limit. Otherwise, the series is said to be divergent.

A classic example is the Basel problem, where \(a_n = 1/n^2\). Then \[\sum_{n = 1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}.\]

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 \(\sum_{n = 1}^\infty a_n\) does not discriminate between different orderings of the sequence \(\{a_n\}\), while 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. This is defined as follows. A series is absolutely convergent if \(\sum_{n = 1}^\infty |a_n|\) is well defined. Furthermore, all possible orderings give the same value.

Otherwise, the series is conditionally convergent. 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 \(\pm \infty\).

The definition of continuity at a point is given through limits.

The above definition of a limit is true even if \(f(c) \neq L\). Indeed, the function f need not even be defined at c. However, if \(f(c)\) is defined and is equal to \(L\), then the function is said to be continuous at the point \(c\).

Equivalently, the function is continuous at \(c\) if \(f(x) \rightarrow f(c)\) as \(x \rightarrow c\), or in terms of sequences, whenever \(x_n \rightarrow c\), then \(f(x_n) \rightarrow f(c)\).

An example of a limit where \(f\) is not defined at \(c\) is given below.

Consider the function

\[f(x) = \frac{x^2 - 1}{x - 1}.\]

then f(1) is not defined (see Indeterminate form), yet as x moves arbitrarily close to 1, f(x) correspondingly approaches 2:

• f(100) = 1.9900
• f(1000) = 1.9990
• f(10000) = 1.9999

Condensed: the full section is in Wikipedia.

An important class of functions when considering limits are continuous functions. These are precisely those functions which preserve limits, in the sense that if \(f\) is a continuous function, then whenever \(a_n \rightarrow a\) in the domain of \(f\), then the limit \(f(a_n)\) exists and furthermore is \(f(a)\).

In the most general setting of topological spaces, a short proof is given below:

Let \(f: X\rightarrow Y\) be a continuous function between topological spaces \(X\) and \(Y\). By definition, for each open set \(V\) in \(Y\), the preimage \(f^{-1}(V)\) is open in \(X\).

Now suppose \(a_n \rightarrow a\) is a sequence with limit \(a\) in \(X\). Then \(f(a_n)\) is a sequence in \(Y\), and \(f(a)\) is some point.

Choose a neighborhood \(V\) of \(f(a)\). Then \(f^{-1}(V)\) is an open set (by continuity of \(f\)) which in particular contains \(a\), and therefore \(f^{-1}(V)\) is a neighborhood of \(a\). By the convergence of \(a_n\) to \(a\), there exists an \(N\) such that for \(n > N\), we have \(a_n \in f^{-1}(V)\).

Then applying \(f\) to both sides gives that, for the same \(N\), for each \(n > N\) we have \(f(a_n) \in V\). Originally \(V\) was an arbitrary neighborhood of \(f(a)\), so \(f(a_n) \rightarrow f(a)\). This concludes the proof.

In real analysis, for the more concrete case of real-valued functions defined on a subset \(E \subset \mathbb{R}\), that is, \(f: E \rightarrow \mathbb{R}\), a continuous function may also be defined as a function which is continuous at every point of its domain.

In topology, limits are used to define limit points of a subset of a topological space, which in turn give a useful characterization of closed sets.

In a topological space \(X\), consider a subset \(S\). A point \(a\) is called a limit point if there is a sequence \(\{a_n\}\) in \(S\setminus\{a\}\) such that \(a_n \rightarrow a\).

The reason why \(\{a_n\}\) is defined to be in \(S\setminus\{a\}\) rather than just \(S\) is illustrated by the following example. Take \(X = \mathbb{R}\) and \(S = [0,1] \cup \{2\}\). Then \(2 \in S\), and therefore is the limit of the constant sequence \(2, 2, \cdots\). But \(2\) is not a limit point of \(S\).

A closed set, which is defined to be the complement of an open set, is equivalently any set \(C\) which contains all its limit points.

The derivative is defined formally as a limit. In the scope of real analysis, the derivative is first defined for real functions \(f\) defined on a subset \(E \subset \mathbb{R}\). The derivative at \(x \in E\) is defined as follows. If the limit of \[\frac{f(x+h) - f(x)}{h}\] as \(h \rightarrow 0\) exists, then the derivative at \(x\) is this limit.

Equivalently, it is the limit as \(y \rightarrow x\) of \[\frac{f(y) - f(x)}{y-x}.\]

If the derivative exists, it is commonly denoted by \(f'(x)\).

For sequences of real numbers, a number of properties can be proven. Suppose \(\{a_n\}\) and \(\{b_n\}\) are two sequences converging to \(a\) and \(b\) respectively.

• Sum of limits is equal to limit of sum\[a_n + b_n \rightarrow a + b.\]
• Product of limits is equal to limit of product\[a_n \cdot b_n \rightarrow a \cdot b.\]
• Inverse of limit is equal to limit of inverse (as long as \(a \neq 0\))\[\frac{1}{a_n} \rightarrow \frac{1}{a}.\]

Equivalently, the function \(f(x) = 1/x\) is continuous about nonzero \(x\).

Beyond whether or not a sequence \(\{a_n\}\) converges to a limit \(a\), it is possible to describe how fast a sequence converges to a limit. One way to quantify this is using the order of convergence of a sequence.

A formal definition of order of convergence can be stated as follows. Suppose \(\{a_n\}_{n > 0}\) is a sequence of real numbers which is convergent with limit \(a\). Furthermore, \(a_n \neq a\) for all \(n\). If positive constants \(\lambda\) and \(\alpha\) exist such that \[\lim_{n \to \infty } \frac{ \left| a_{n+1} - a \right| }{ \left| a_n - a \right| ^\alpha } = \lambda\] then \(a_n\) is said to converge to \(a\) with order of convergence \(\alpha\). The constant \(\lambda\) is known as the asymptotic error constant.

Order of convergence is used for example the field of numerical analysis, in error analysis.

Limits can be difficult to compute. There exist limit expressions whose modulus of convergence is undecidable. In recursion theory, the limit lemma proves that it is possible to encode undecidable problems using limits.

There are several theorems or tests that indicate whether the limit exists. These are known as convergence tests. Examples include the ratio test and the squeeze theorem. However they may not tell how to compute the limit.