In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the lim {\displaystyle \lim }...

topology and related branches, a net or Moore–Smith sequence is a function whose domain is a directed set. The codomain of this function is usually some...

mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can...

definition of a cluster or accumulation point of a sequence generalizes to nets and filters. The similarly named notion of a limit point of a sequence (respectively...

mathematics, a limit is the value that a function (or sequence) approaches as the argument (or index) approaches some value. Limits of functions are...

mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input...

In mathematics, an ultralimit is a geometric construction that assigns a limit metric space to a sequence of metric spaces X n {\displaystyle X_{n}} ...

mathematics, the limit of a sequence of sets A 1 , A 2 , … {\displaystyle A_{1},A_{2},\ldots } (subsets of a common set X {\displaystyle X} ) is a set whose...

is the limit of the sequence (3, 3.1, 3.14, 3.141, 3.1415, ...), which is increasing. A related sequence is the sequence of decimal digits of π, that...

is that a uniformly convergent sequence of continuous functions has a continuous limit. The failure of a merely pointwise-convergent limit of continuous...

as a continued fraction. Using calculus, e may also be represented as an infinite series, infinite product, or other types of limit of a sequence. Euler...

In mathematics, a subsequential limit of a sequence is the limit of some subsequence. Every subsequential limit is a cluster point, but not conversely...

of the sequence are getting closer and closer together in a way that suggests that the sequence ought to have a limit in X. Nonetheless, such a limit...

term of the sequence. In this sense, a limit ordinal is the limit of all smaller ordinals (indexed by itself). Put more directly, it is the supremum of the...

specifically in the theory of generalized functions, the limit of a sequence of distributions is the distribution that sequence approaches. The distance...

}a_{i}=\lim _{n\to \infty }\,\sum _{i=1}^{n}a_{i},} if it exists. When the limit exists, the series is convergent or summable and also the sequence (...

14159. One method of deriving this formula, which originated with Archimedes, involves viewing the circle as the limit of a sequence of regular polygons...

often constructed as the limit of a sequence of nested intersections, or (more generally) as the inverse limit of a sequence of continua. The buckethandle...

{\displaystyle {\begin{aligned}b&=a{\sqrt {1-e^{2}}},\\a\ell &=b^{2}.\end{aligned}}} A parabola can be obtained as the limit of a sequence of ellipses where one focus...

gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence...

_δ)-definition of limit, formal definition of the mathematical notion of limit Limit of a sequence One-sided limit, either of the two limits of a function as a specified...

study of dense graphs. Graphons arise both as a natural notion for the limit of a sequence of dense graphs, and as the fundamental defining objects of exchangeable...

In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the...

central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample mean converges to a standard...

arithmetic–geometric mean (AGM or agM) of two positive real numbers x and y is the mutual limit of a sequence of arithmetic means and a sequence of geometric means. The...

(and only if) the limit of the sequence f n ( x ) {\displaystyle f_{n}(x)} evaluated at each point x {\displaystyle x} in the domain of f {\displaystyle...

geometry, a horosphere (or parasphere) is a specific hypersurface in hyperbolic n-space. It is the boundary of a horoball, the limit of a sequence of increasing...

strings, or as the limit of a sequence of strings, it can be used to study a wide variety of mathematical objects, including integers. One of the main motivations...

particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability...

when a point is the limit of a sequence. Still, for some spaces that are too large in some sense, one specifies also when a point is the limit of more...