In real analysis for functions of one variable, a limit definition "Definition (mathematics)") can be made similar to that of the limit of a sequence, in which the values that the function takes within an interval "Interval (mathematics)") or radius of convergence approach a fixed point c — accumulation point —, regardless of whether this belongs to the domain of the function.[10] This can be further generalized to functions of several variables or functions in different spaces. metric.

Colloquially, it is said that the limit of the function f(x) when x tends to c is L, and it is written:.

if one can find for each occasion an x sufficiently close to c such that the value of f(x) is as close to L as desired.

For greater mathematical rigor, the epsilon-delta definition of limit is used, which is stricter and makes the limit a great tool for real analysis. Its definition is the following:

This definition can be written using logical-mathematical terms and in a compact way:.

This definition is equivalent to the limit of a sequence, a function is continuous if:.

In set theory, the concept of limit is also used, which can be calculated on a sequence of sets. To do this, the sets must meet a series of conditions, such as monotony (increasing or decreasing). More generally, and using the definition of upper limit and lower limit for any sequence of sets, it is said that the limit of this sequence exists if the upper limit and lower limit exist and are equal. In general you have:

If the first term limit and the penultimate term are equal, then all equalities are verified. These concepts are very useful in mathematical disciplines such as measure theory, especially in probability spaces. It is not difficult to construct non-convergent sequences where it is verified that:

Networks

All the above notions of limit can be unified and generalized to arbitrary topological spaces by introducing topological networks and defining their limits.

Let be a topological space and a network in . It is said to be a limit point of the network if the network is finally in every environment "Environment (mathematics)") of , that is, if whatever the environment of (that is, whatever the set such that there exists an open such that ) there exists such that for each with it holds that .

Filters

In the case of "Filter (mathematics)" filters, since they are mathematical objects similar to topological networks, the definition of a limit is also possible. Indeed, let X be a topological space and x be a point of X. A base filter B is said to converge to x, denoted as B → x or , if for every environment "Environment (mathematics)") U of x, there exists a B ∈ B such that B ⊆ U. In this case, x is called the limit of B and B is called the convergent base filter.[11]​[12]​.

In the same way, it can be applied to functions, extending the definition of continuity to them. If X, Y are two topological spaces and f: X → Y is a function, with B being a neighborhood filter on X of a point a belonging to X, then the limit with respect to the filter B of f is y, denoted as.

if B converges to a, then f converges to y; In other words, y is the limit of f at the point a.[11]​.

In functional analysis, a Banach limit is a continuous linear functional defined on the Banach space for any bounded sequence of complex numbers, where a series of conditions are met, among which it is found that if it is a convergent sequence, then , generalizing the concept of limit. Therefore, it is an extension of the continuous functional [13]​.

In particular, the existence of the Banach limit is not unique.[13]​.

In category theory, a branch of mathematics, the abstract concept of limit is defined, which uses essential properties of universal constructions such as products "Product (category theory)") and inverse limits.

Infinity as a limit

There is also the notion that a limit "tends to infinity", rather than to a finite value. A sequence is said to "tend to infinity" if, for each real number, known as the boundary, there exists an integer such that for each,

That is, for every possible boundary, the succession will exceed the boundary. This is often expressed as follows or simply:

It is possible for a sequence to be divergent, but not tend to infinity. Such sequences are called oscillatory. For example, an oscillatory sequence is .

There is a corresponding notion of tendency to negative infinity, , defined by modifying the inequality in the previous definition to with.

A sequence with is called boundaryless, a definition also valid for sequences of complex numbers, or in any metric space. Sequences that do not tend to infinity are called bounded. Sequences that do not tend to a positive infinity are called bounded above, while those that do not tend to a negative infinity are called bounded below.

This notion is used in dynamic systems, to study the limits of trajectories. Defining a trajectory as a function, the point defines the "position" of the trajectory at "time". The limit set of a trajectory is defined as follows. For any sequence of increasing times, there is an associated sequence of positions. If it is the limit of the sequence for any sequence of increasing times, then it is a limit set of the trajectory.

Technically, this is the limit set. The corresponding set of limits for sequences of decreasing time is called the limit set.

An illustrative example is the circular path: . It does not have a unique limit, but for each , the point is a limit point, given the succession of times . But it is not necessary that the limit points be reached by the trajectory. The trajectory also has the unit circle as its limit set.

The concept of limit of a sequence "Sequence (mathematical)") refers to an element to which the terms of the sequence are progressively approached. For this value to exist, there must be a term from which the sequence is "trapped" within an open ball of arbitrarily small radius centered on said value.

Let be a sequence of elements in a metric space. It is said to be the limit of the sequence and is denoted as:.

if, and only if, for every real value, a natural number can be found such that all terms of the sequence starting from are at a distance of less than . Written in formal language:[8]​.

If there is a value that satisfies the previous condition, then this value is unique.[9]​ In this case, the sequence is said to be convergent, since it converges or tends to when it tends to infinity. Otherwise, the sequence has no limit and is said to be divergent.

In the case of real numbers, the distance is given by the absolute value of the difference. The long-term behavior (for ) of divergent sequences of real numbers can vary: there are sequences that tend toward infinity (positive or negative), that oscillate within a range (delimited by an upper limit and a lower limit), or none of the above.

In real analysis for functions of one variable, a limit definition "Definition (mathematics)") can be made similar to that of the limit of a sequence, in which the values that the function takes within an interval "Interval (mathematics)") or radius of convergence approach a fixed point c — accumulation point —, regardless of whether this belongs to the domain of the function.[10] This can be further generalized to functions of several variables or functions in different spaces. metric.

Colloquially, it is said that the limit of the function f(x) when x tends to c is L, and it is written:.

if one can find for each occasion an x sufficiently close to c such that the value of f(x) is as close to L as desired.

For greater mathematical rigor, the epsilon-delta definition of limit is used, which is stricter and makes the limit a great tool for real analysis. Its definition is the following:

This definition can be written using logical-mathematical terms and in a compact way:.

This definition is equivalent to the limit of a sequence, a function is continuous if:.

In set theory, the concept of limit is also used, which can be calculated on a sequence of sets. To do this, the sets must meet a series of conditions, such as monotony (increasing or decreasing). More generally, and using the definition of upper limit and lower limit for any sequence of sets, it is said that the limit of this sequence exists if the upper limit and lower limit exist and are equal. In general you have:

If the first term limit and the penultimate term are equal, then all equalities are verified. These concepts are very useful in mathematical disciplines such as measure theory, especially in probability spaces. It is not difficult to construct non-convergent sequences where it is verified that:

Networks

All the above notions of limit can be unified and generalized to arbitrary topological spaces by introducing topological networks and defining their limits.

Let be a topological space and a network in . It is said to be a limit point of the network if the network is finally in every environment "Environment (mathematics)") of , that is, if whatever the environment of (that is, whatever the set such that there exists an open such that ) there exists such that for each with it holds that .

Filters

In the case of "Filter (mathematics)" filters, since they are mathematical objects similar to topological networks, the definition of a limit is also possible. Indeed, let X be a topological space and x be a point of X. A base filter B is said to converge to x, denoted as B → x or , if for every environment "Environment (mathematics)") U of x, there exists a B ∈ B such that B ⊆ U. In this case, x is called the limit of B and B is called the convergent base filter.[11]​[12]​.

In the same way, it can be applied to functions, extending the definition of continuity to them. If X, Y are two topological spaces and f: X → Y is a function, with B being a neighborhood filter on X of a point a belonging to X, then the limit with respect to the filter B of f is y, denoted as.

if B converges to a, then f converges to y; In other words, y is the limit of f at the point a.[11]​.

In functional analysis, a Banach limit is a continuous linear functional defined on the Banach space for any bounded sequence of complex numbers, where a series of conditions are met, among which it is found that if it is a convergent sequence, then , generalizing the concept of limit. Therefore, it is an extension of the continuous functional [13]​.

In particular, the existence of the Banach limit is not unique.[13]​.

In category theory, a branch of mathematics, the abstract concept of limit is defined, which uses essential properties of universal constructions such as products "Product (category theory)") and inverse limits.

Infinity as a limit

There is also the notion that a limit "tends to infinity", rather than to a finite value. A sequence is said to "tend to infinity" if, for each real number, known as the boundary, there exists an integer such that for each,

That is, for every possible boundary, the succession will exceed the boundary. This is often expressed as follows or simply:

It is possible for a sequence to be divergent, but not tend to infinity. Such sequences are called oscillatory. For example, an oscillatory sequence is .

There is a corresponding notion of tendency to negative infinity, , defined by modifying the inequality in the previous definition to with.

A sequence with is called boundaryless, a definition also valid for sequences of complex numbers, or in any metric space. Sequences that do not tend to infinity are called bounded. Sequences that do not tend to a positive infinity are called bounded above, while those that do not tend to a negative infinity are called bounded below.

This notion is used in dynamic systems, to study the limits of trajectories. Defining a trajectory as a function, the point defines the "position" of the trajectory at "time". The limit set of a trajectory is defined as follows. For any sequence of increasing times, there is an associated sequence of positions. If it is the limit of the sequence for any sequence of increasing times, then it is a limit set of the trajectory.

Technically, this is the limit set. The corresponding set of limits for sequences of decreasing time is called the limit set.

An illustrative example is the circular path: . It does not have a unique limit, but for each , the point is a limit point, given the succession of times . But it is not necessary that the limit points be reached by the trajectory. The trajectory also has the unit circle as its limit set.