limit (mathematics)

TheInfoList

OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a limit is the value that a function (or
sequence In mathematics, a sequence is an enumerated collection of mathematical object, objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''eleme ...
) approaches as the input (or index) approaches some value. Limits are essential to
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
and
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, and are used to define continuity,
derivative In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
s, and
integral In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
s. The concept of a
limit of a sequence As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1." In mathematics, the limit ...
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 Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...
. In formulas, a limit of a function is usually written as :$\lim_ f\left(x\right) = L,$ (although a few authors may use "Lt" instead of "lim") and is read as "the limit of of as approaches equals ". The fact that a function approaches the limit as approaches is sometimes denoted by a right arrow (→ or $\rightarrow$), as in :$f\left(x\right) \to L \text x \to c,$ which reads "$f$ of $x$ tends to $L$ as $x$ tends to $c$".

# History

Grégoire de Saint-Vincent gave the first definition of limit (terminus) of a
geometric series In mathematics, a geometric series is the sum of an infinite number of Summand, terms that have a constant ratio between successive terms. For example, 1/2 + 1/4 + 1/8 + 1/16 + · · ·, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \, ...
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." The modern definition of a limit goes back to
Bernard Bolzano Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Gonzal Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typica ...
who, in 1817, introduced the basics of the epsilon-delta technique to define continuous functions. However, his work was not known during his lifetime. Augustin-Louis Cauchy in 1821, followed by
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university without a degree, ...
, 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 is due to G. H. Hardy, who introduced it in his book '' A Course of Pure Mathematics'' in 1908.

# Types of limits

## In sequences

### Real numbers

The expression 0.999... should be interpreted as the limit of the sequence 0.9, 0.99, 0.999, ... and so on. This sequence can be rigorously shown to have the limit 1, and therefore this expression is meaningfully interpreted as having the value 1. Formally, suppose is a
sequence In mathematics, a sequence is an enumerated collection of mathematical object, objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''eleme ...
of
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s. When the limit of the sequence exists, the real number is the ''limit'' of this sequence if and only if for every
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
, there exists a
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
such that for all , we have . The notation $\lim_ a_n = L$ is often used, and which is read as :"The limit of ''an'' as ''n'' approaches infinity equals ''L''" The formal definition intuitively means that eventually, all elements of the sequence get arbitrarily close to the limit, since the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign (mathematics), sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative number, negative (in which cas ...
is the distance between and . Not every sequence has a limit. If it does, then it is called '' convergent'', and if it does not, then it is ''divergent''. One can show that a convergent sequence has only one limit. The limit of a sequence and the limit of a function are closely related. On one hand, the limit as approaches infinity of a sequence is simply the limit at infinity of a function —defined on the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s . On the other hand, if is the domain of a function and if the limit as approaches infinity of is for ''every'' arbitrary sequence of points in which converges to , then the limit of the function as approaches is . One such sequence would be .

### Infinity as a limit

There is also a notion of having a limit "at infinity", as opposed to at some finite $L$. A sequence $\$ is said to "tend to infinity" if, for each real number $M > 0$, known as the bound, there exists an integer $N$ such that for each $n > N$, $, a_n, > M.$ That is, for every possible bound, the magnitude of the sequence eventually exceeds the bound. This is often written $\lim_ a_n = \infty$ or simply $a_n \rightarrow \infty$. Such sequences are also called unbounded. 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 $a_n = \left(-1\right)^n$. For the real numbers, there are corresponding notions of tending to positive infinity and negative infinity, by removing the modulus sign from the above definition: $a_n > M.$ defines tending to positive infinity, while $-a_n > M.$ defines tending to negative infinity. Sequences which do not tend to infinity are called bounded. Sequences which do not tend to positive infinity are called bounded above, while those which do not tend to negative infinity are bounded below.

### Metric space

The discussion of sequences above is for sequences of real numbers. The notion of limits can be defined for sequences valued in more abstract spaces. One example of a more abstract space is
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
s. If $M$ is a metric space with distance function $d$, and $\_$ is a sequence in $M$, then the limit (when it exists) of the sequence is an element $a\in M$ such that, given $\epsilon > 0$, there exists an $N$ such that for each $n > N$, the equation $d(a, a_n) < \epsilon$ is satisfied. An equivalent statement is that $a_n \rightarrow a$ if the sequence of real numbers $d\left(a, a_n\right) \rightarrow 0$.

### = Example: ℝn

= An important example is the space of $n$-dimensional real vectors, with elements $\mathbf = \left(x_1, \cdots, x_n\right)$ where each of the $x_i$ are real, an example of a suitable distance function is the
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two Point (geometry), points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theo ...
, defined by $d(\mathbf, \mathbf) = , \mathbf - \mathbf, = \sqrt.$ The sequence of points $\_$ converges to $\mathbf$ if the limit exists and $, \mathbf_n - \mathbf, \rightarrow 0$.

### Topological space

In some sense the ''most'' abstract space in which limits can be defined are
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s. If $X$ is a topological space with topology $\tau$, and $\_$ is a sequence in $X$, then the limit (when it exists) of the sequence is a point $a\in X$ such that, given a (open)
neighborhood A neighbourhood (British English British English (BrE, en-GB, or BE) is, according to Oxford Dictionaries, " English as used in Great Britain, as distinct from that used elsewhere". More narrowly, it can refer specifically to the En ...
$U\in \tau$ of $a$, there exists an $N$ such that for every $n > N$, $a_n \in U$ is satisfied.

### Function space

This section deals with the idea of limits of sequences of functions, not to be confused with the idea of limits of functions, discussed below. The field of
functional analysis Functional analysis is a branch of mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, ...
partly seeks to identify useful notions of convergence on function spaces. For example, consider the space of functions from a generic set $E$ to $\mathbb$. Given a sequence of functions $\_$ such that each is a function $f_n: E \rightarrow \mathbb$, suppose that there exists a function such that for each $x \in E$, $f_n(x) \rightarrow f(x) \text \lim_f_n(x) = f(x).$ Then the sequence $f_n$ is said to converge pointwise to $f$. However, such sequences can exhibit unexpected behavior. For example, it is possible to construct a sequence of continuous functions which has a discontinuous pointwise limit. Another notion of convergence is
uniform convergence In the mathematics, mathematical field of mathematical analysis, analysis, uniform convergence is a Modes of convergence, mode of Limit of a sequence, convergence of functions stronger than pointwise convergence. A sequence of Function (mathematics ...
. The uniform distance between two functions $f,g: E \rightarrow \mathbb$ is the maximum difference between the two functions as the argument $x \in E$ is varied. That is, $d(f,g) = \max_, f(x) - g(x), .$ Then the sequence $f_n$ is said to uniformly converge or have a uniform limit of $f$ if $f_n \rightarrow f$ with respect to this distance. The uniform limit has "nicer" properties than the pointwise limit. For example, the uniform limit of a sequence of continuous functions is continuous. Many different notions of convergence can be defined on function spaces. This is sometimes dependent on the regularity of the space. Prominent examples of function spaces with some notion of convergence are
Lp space In mathematics, the spaces are function spaces defined using a natural generalization of the Norm (mathematics)#p-norm, -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although ...
s and
Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a normed space, norm that is a combination of Lp norm, ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a ...
.

## In functions

Suppose is a real-valued function and is a
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
. Intuitively speaking, the expression :$\lim_f\left(x\right) = L$ means that can be made to be as close to as desired, by making sufficiently close to . In that case, the above equation can be read as "the limit of of , as approaches , is ". Formally, the definition of the "limit of $f\left(x\right)$ as $x$ approaches $c$" is given as follows. The limit is a real number $L$ so that, given an arbitrary real number $\epsilon > 0$ (thought of as the "error"), there is a $\delta > 0$ such that, for any $x$ satisfying $0 < , x - c, < \delta$, it holds that $, f\left(x\right) - L , < \epsilon$. 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 $\$ in the domain of $f$, there is an associated sequence $\$, 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 $f\left(x_n\right) \rightarrow L$.

### One-sided limit

It is possible to define the notion of having a limit "from above" or "left limit", and a notion of a limit "from below" or "right limit". These need not agree. An example is given by the positive
indicator function In mathematics, an indicator function or a characteristic function of a subset of a Set (mathematics), set is a Function (mathematics), function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset ...
, $f: \mathbb \rightarrow \mathbb$, defined such that $f\left(x\right) = 0$ if $x \leq 0$, and $f\left(x\right) = 1$ if $x > 0$. At $x = 0$, the function has a "left limit" of 0, a "right limit" of 1, and its limit does not exist.

### Infinity in limits of functions

It is possible to define the notion of "tending to infinity" in the domain of $f$, $\lim_ f(x) = L.$ In this expression, the infinity is considered to be signed: either $+ \infty$ or $- \infty$. The "limit of f as x tends to positive infinity" is defined as follows. It is a real number $L$ such that, given any real $\epsilon > 0$, there exists an $M > 0$ so that if $x > M$, $, f\left(x\right) - L, < \epsilon$. Equivalently, for any sequence $x_n \rightarrow + \infty$, we have $f\left(x_n\right) \rightarrow L$. It is also possible to define the notion of "tending to infinity" in the value of $f$, $\lim_ f(x) = \infty.$ The definition is given as follows. Given any real number $M>0$, there is a $\delta > 0$ so that for $0 < , x - c, < \delta$, the absolute value of the function $, f\left(x\right), > M$. Equivalently, for any sequence $x_n \rightarrow c$, the sequence $f\left(x_n\right) \rightarrow \infty$.

## Nonstandard analysis

In non-standard analysis (which involves a hyperreal enlargement of the number system), the limit of a sequence $\left(a_n\right)$ 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, :$\lim_ a_n = \operatorname\left(a_H\right) .$ Here, the standard part function "st" rounds off each finite hyperreal number to the nearest real number (the difference between them is
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to 0, zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century New Latin, Modern Latin coinage ''infinitesimus'', which ori ...
). 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

## Limit sets

### Limit set of a sequence

Let $\_$ be a sequence in a topological space $X$. For concreteness, $X$ can be thought of as $\mathbb$, but the definitions hold more generally. The
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 ca ...
is the set of points such that if there is a convergent
subsequence In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a ...
$\_$ with $a_\rightarrow a$, then $a$ belongs to the limit set. In this context, such an $a$ is sometimes called a limit point. A use of this notion is to characterize the "long-term behavior" of oscillatory sequences. For example, consider the sequence $a_n = \left(-1\right)^n$. Starting from n=1, the first few terms of this sequence are $-1, +1, -1, +1, \cdots$. It can be checked that it is oscillatory, so has no limit, but has limit points $\$.

### Limit set of a trajectory

This notion is used in
dynamical systems In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
, to study limits of trajectories. Defining a trajectory to be a function $\gamma: \mathbb \rightarrow X$, the point $\gamma\left(t\right)$ 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 $\$, there is an associated sequence of positions $\ = \$. If $x$ is the limit set of the sequence $\$ for any sequence of increasing times, then $x$ is a limit set of the trajectory. Technically, this is the $\omega$-limit set. The corresponding limit set for sequences of decreasing time is called the $\alpha$-limit set. An illustrative example is the circle trajectory: $\gamma\left(t\right) = \left(\cos\left(t\right), \sin\left(t\right)\right)$. This has no unique limit, but for each $\theta \in \mathbb$, the point $\left(\cos\left(\theta\right), \sin\left(\theta\right)\right)$ is a limit point, given by the sequence of times $t_n = \theta + 2\pi n$. But the limit points need not be attained on the trajectory. The trajectory $\gamma\left(t\right) = t/\left(1 + t\right)\left(\cos\left(t\right), \sin\left(t\right)\right)$ also has the unit circle as its limit set.

# Uses

Limits are used to define a number of important concepts in analysis.

## Series

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 $\sum_^\infty a_n.$ This is defined through limits as follows: given a sequence of real numbers $\$, the sequence of partial sums is defined by $s_n = \sum_^n a_i.$ If the limit of the sequence $\$ exists, the value of the expression $\sum_^\infty a_n$ is defined to be the limit. Otherwise, the series is said to be divergent. A classic example is the
Basel problem The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 ...
, where $a_n = 1/n^2$. Then $\sum_^\infty \frac = \frac.$ 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 $\sum_^\infty a_n$ does not discriminate between different orderings of the sequence $\$, 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 $\sum_^\infty , a_n,$ 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 $\pm \infty$.

### Power series

A useful application of the theory of sums of series is for power series. These are sums of series of the form $f(z) = \sum_^\infty c_n z^n.$ Often $z$ is thought of as a complex number, and a suitable notion of convergence of complex sequences is needed. The set of values of $z\in \mathbb$ for which the series sum converges is a circle, with its radius known as the radius of convergence.

## Continuity of a function at a point

The definition of continuity at a point is given through limits. The above definition of a limit is true even if $f\left(c\right) \neq L$. Indeed, the function need not even be defined at . However, if $f\left(c\right)$ 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 $f\left(x\right) \rightarrow f\left(c\right)$ as $x \rightarrow c$, or in terms of sequences, whenever $x_n \rightarrow c$, then $f\left(x_n\right) \rightarrow f\left(c\right)$. An example of a limit where $f$ is not defined at $c$ is given below. Consider the function $f(x) = \frac.$ then is not defined (see Indeterminate form), yet as moves arbitrarily close to 1, correspondingly approaches 2: Thus, can be made arbitrarily close to the limit of 2—just by making sufficiently close to . In other words, $\lim_ \frac = 2.$ This can also be calculated algebraically, as $\frac = \frac = x+1$ for all real numbers . Now, since is continuous in at 1, we can now plug in 1 for , leading to the equation $\lim_ \frac = 1+1 = 2.$ In addition to limits at finite values, functions can also have limits at infinity. For example, consider the function $f(x) = \frac$ where: * * * As becomes extremely large, the value of approaches , and the value of can be made as close to as one could wish—by making sufficiently large. So in this case, the limit of as approaches infinity is , or in mathematical notation,$\lim_\frac = 2.$

## Continuous functions

An important class of functions when considering limits are
continuous function In mathematics, a continuous function is a function (mathematics), function such that a continuous variation (that is a change without jump) of the argument of a function, argument induces a continuous variation of the Value (mathematics), value ...
s. 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 $f\left(a_n\right)$ exists and furthermore is $f\left(a\right)$. 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 $f^\left(V\right)$ is open in $X$. Now suppose $a_n \rightarrow a$ is a sequence with limit $a$ in $X$. Then $f\left(a_n\right)$ is a sequence in $Y$, and $f\left(a\right)$ is some point. Choose a neighborhood $V$ of $f\left(a\right)$. Then $f^\left(V\right)$ is an open set (by continuity of $f$) which in particular contains $a$, and therefore $f^\left(V\right)$ is a neighborhood of $a$. By the convergence of $a_n$ to $a$, there exists an $N$ such that for $n > N$, we have $a_n \in f^\left(V\right)$. Then applying $f$ to both sides gives that, for the same $N$, for each $n > N$ we have $f\left(a_n\right) \in V$. Originally $V$ was an arbitrary neighborhood of $f\left(a\right)$, so $f\left(a_n\right) \rightarrow f\left(a\right)$. This concludes the proof. In real analysis, for the more concrete case of real-valued functions defined on a subset $E \subset \mathbb$, that is, $f: E \rightarrow \mathbb$, a continuous function may also be defined as a function which is continuous at every point of its domain.

## Limit points

In
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, limits are used to define
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined ...
s of a subset of a topological space, which in turn give a useful characterization of
closed set In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...
s. In a topological space $X$, consider a subset $S$. A point $a$ is called a limit point if there is a sequence $\$ in $S\backslash\$ such that $a_n \rightarrow a$. The reason why $\$ is defined to be in $S\backslash\$ rather than just $S$ is illustrated by the following example. Take $X = \mathbb$ and . 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.

## Derivative

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 $E \subset \mathbb$. The derivative at $x \in E$ is defined as follows. If the limit $\frac$ as $h \rightarrow 0$ exists, then the derivative at $x$ is this limit. Equivalently, it is the limit as $y \rightarrow x$ of $\frac.$ If the derivative exists, it is commonly denoted by $f\text{'}\left(x\right)$.

# Properties

## Sequences of real numbers

For sequences of real numbers, a number of properties can be proven. Suppose $\$ and $\$ 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 $a \neq 0$) $\frac \rightarrow \frac.$ equivalently, the function $f\left(x\right) = 1/x$ is continuous about positive $x$.

### Cauchy sequences

A property of convergent sequences of real numbers is that they are Cauchy sequences. The definition of a Cauchy sequence $\$ is that for every real number $\epsilon > 0$, there is an $N$ such that whenever $m, n > N$, $, a_m - a_n, < \epsilon.$ 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. Cauchy sequences are closely related to convergent sequences. In fact, for sequences of real numbers they are equivalent: any Cauchy sequence is convergent. 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 In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
, $\mathbb$, with the usual distance. The sequence of decimal approximations to $\sqrt$, truncated at the $n$th decimal place is a Cauchy sequence, but ''does not converge in'' $\mathbb$. A metric space in which every Cauchy sequence is also convergent, that is, Cauchy sequences are equivalent to convergent sequences, is known as a
complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence#In a metric space, Cauchy sequence of points in has a Limit of a sequence, limit that is also in . Intuitively, a space is complete if ther ...
. 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 $a$.

## Order of convergence

Beyond whether or not a sequence $\$ 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 $\_$ 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 $\lim_ \frac = \lambda$ 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 Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
, in error analysis.

## Computability

Limits can be difficult to compute. There exist limit expressions whose modulus of convergence is undecidable. In
recursion theory Computability theory, also known as recursion theory, is a branch of mathematical logic Mathematical logic is the study of formal logic within mathematics Mathematics is an area of knowledge that includes the topics of numbers, formu ...
, 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 In mathematics, convergence tests are methods of testing for the Convergent series, convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an series (mathematics), infinite series \sum_^\infty a_n. L ...
. Examples include the ratio test and the squeeze theorem. However they may not tell how to compute the limit.

*
Asymptotic analysis In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing Limit (mathematics), limiting behavior. As an illustration, suppose that we are interested in the properties of a function as becomes very larg ...
: a method of describing limiting behavior **
Big O notation Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a memb ...
: used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity *
Banach limit In mathematical analysis, a Banach limit is a Continuous function, continuous linear functional \phi: \ell^\infty \to \mathbb defined on the Banach space \ell^\infty of all Bounded function, bounded Complex number, complex-valued sequences such tha ...
defined on the Banach space $\ell^\infty$ that extends the usual limits. * Convergence of random variables * Convergent matrix * Limit in category theory **
Direct limit In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be Group (mathematics), groups, Ring (mathematics), rings, Vector spac ...
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Inverse limit In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related mathematical object, objects, the precise gluing process being specified by morphisms between the objects. Thu ...
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Limit of a function 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. In other words, the limit of (sin ''x'')/''x'', as ''x'' approaches z ...
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One-sided limit In calculus, a one-sided limit refers to either one of the two Limit of a function, limits of a Function (mathematics), function f(x) of a Real number, real variable x as x approaches a specified point either from the left or from the right. The ...
: either of the two limits of functions of a real variable ''x'', as ''x'' approaches a point from above or below ** List of limits: list of limits for common functions ** Squeeze theorem: finds a limit of a function via comparison with two other functions * Limit superior and limit inferior * Modes of convergence ** An annotated index

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