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Limit (mathematics), the Glossary

In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value.^[1]

87 relations: A Course of Pure Mathematics, Absolute convergence, Absolute value, Accumulation point, Addison-Wesley, Asymptotic analysis, Augustin-Louis Cauchy, Banach limit, Basel problem, Bernard Bolzano, Big O notation, Calculus, Category theory, Cauchy sequence, Cengage Group, Closed set, Complete metric space, Complex number, Computability theory, Computation in the limit, Conditional convergence, Continuous function, Convergence of random variables, Convergence tests, Convergent matrix, Convergent series, Derivative, Direct limit, Dynamical system, Euclid's Elements, Euclidean distance, Function (mathematics), Functional analysis, G. H. Hardy, Geometric series, Grégoire de Saint-Vincent, Hausdorff space, Hermann Hankel, Hyperinteger, Hyperreal number, Indeterminate form, Indicator function, Infinitesimal, Integral, Inverse limit, Karl Weierstrass, Limit (category theory), Limit inferior and limit superior, Limit of a function, Limit of a sequence, ... Expand index (37 more) »
Convergence (mathematics)
Limits (mathematics)

A Course of Pure Mathematics

A Course of Pure Mathematics is a classic textbook in introductory mathematical analysis, written by G. H. Hardy.

See Limit (mathematics) and A Course of Pure Mathematics

In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. Limit (mathematics) and absolute convergence are convergence (mathematics).

See Limit (mathematics) and Absolute convergence

Absolute value

In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign.

See Limit (mathematics) and Absolute value

Accumulation point

In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x contains a point of S other than x itself. Limit (mathematics) and accumulation point are general topology.

See Limit (mathematics) and Accumulation point

Addison-Wesley

Addison–Wesley is an American publisher of textbooks and computer literature.

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Asymptotic analysis

In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.

See Limit (mathematics) and Asymptotic analysis

Augustin-Louis Cauchy

Baron Augustin-Louis Cauchy (France:,&ensp;; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist.

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Banach limit

In mathematical analysis, a Banach limit is a continuous linear functional \phi: \ell^\infty \to \mathbb defined on the Banach space \ell^\infty of all bounded complex-valued sequences such that for all sequences x.

See Limit (mathematics) and Banach limit

Basel problem

The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares.

See Limit (mathematics) and Basel problem

Bernard Bolzano

Bernard Bolzano (born Bernardus Placidus Johann Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his liberal views.

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Big O notation

Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Limit (mathematics) and Big O notation are asymptotic analysis.

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Calculus

Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

See Limit (mathematics) and Calculus

Category theory

Category theory is a general theory of mathematical structures and their relations.

See Limit (mathematics) and Category theory

Cauchy sequence

In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. Limit (mathematics) and Cauchy sequence are convergence (mathematics).

See Limit (mathematics) and Cauchy sequence

Cengage Group

Cengage Group is an American educational content, technology, and services company for higher education, K–12, professional, and library markets.

See Limit (mathematics) and Cengage Group

Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. Limit (mathematics) and closed set are general topology.

See Limit (mathematics) and Closed set

Complete metric space

In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in.

See Limit (mathematics) and Complete metric space

Complex number

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted, called the imaginary unit and satisfying the equation i^.

See Limit (mathematics) and Complex number

Computability theory

Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees.

See Limit (mathematics) and Computability theory

Computation in the limit

In computability theory, a function is called limit computable if it is the limit of a uniformly computable sequence of functions.

See Limit (mathematics) and Computation in the limit

Conditional convergence

In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely. Limit (mathematics) and conditional convergence are convergence (mathematics).

See Limit (mathematics) and Conditional convergence

Continuous function

In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function.

See Limit (mathematics) and Continuous function

Convergence of random variables

In probability theory, there exist several different notions of convergence of sequences of random variables, including convergence in probability, convergence in distribution, and almost sure convergence. Limit (mathematics) and convergence of random variables are convergence (mathematics).

See Limit (mathematics) and Convergence of random variables

Convergence tests

In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series \sum_^\infty a_n. Limit (mathematics) and convergence tests are convergence (mathematics).

See Limit (mathematics) and Convergence tests

Convergent matrix

In linear algebra, a convergent matrix is a matrix that converges to the zero matrix under matrix exponentiation. Limit (mathematics) and convergent matrix are limits (mathematics).

See Limit (mathematics) and Convergent matrix

Convergent series

In mathematics, a series is the sum of the terms of an infinite sequence of numbers. Limit (mathematics) and Convergent series are convergence (mathematics).

See Limit (mathematics) and Convergent series

Derivative

The derivative is a fundamental tool of calculus that quantifies the sensitivity of change of a function's output with respect to its input. Limit (mathematics) and derivative are differential calculus.

See Limit (mathematics) and Derivative

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.

See Limit (mathematics) and Direct limit

Dynamical system

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve.

See Limit (mathematics) and Dynamical system

Euclid's Elements

The Elements (Στοιχεῖα) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid 300 BC.

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Euclidean distance

In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them.

See Limit (mathematics) and Euclidean distance

Function (mathematics)

In mathematics, a function from a set to a set assigns to each element of exactly one element of.

See Limit (mathematics) and Function (mathematics)

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner product, norm, or topology) and the linear functions defined on these spaces and suitably respecting these structures.

See Limit (mathematics) and Functional analysis

G. H. Hardy

Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis.

See Limit (mathematics) and G. H. Hardy

Geometric series

In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms.

See Limit (mathematics) and Geometric series

Grégoire de Saint-Vincent

Grégoire de Saint-Vincent - in Latin: Gregorius a Sancto Vincentio, in Dutch: Gregorius van St-Vincent - (8 September 1584 Bruges – 5 June 1667 Ghent) was a Flemish Jesuit and mathematician.

See Limit (mathematics) and Grégoire de Saint-Vincent

Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each that are disjoint from each other.

See Limit (mathematics) and Hausdorff space

Hermann Hankel

Hermann Hankel (14 February 1839 – 29 August 1873) was a German mathematician.

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Hyperinteger

In nonstandard analysis, a hyperinteger n is a hyperreal number that is equal to its own integer part.

See Limit (mathematics) and Hyperinteger

Hyperreal number

In mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers.

See Limit (mathematics) and Hyperreal number

Indeterminate form

In calculus, it is usually possible to compute the limit of the sum, difference, product, quotient or power of two functions by taking the corresponding combination of the separate limits of each respective function. Limit (mathematics) and Indeterminate form are limits (mathematics).

See Limit (mathematics) and Indeterminate form

Indicator function

In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. Limit (mathematics) and indicator function are real analysis.

See Limit (mathematics) and Indicator function

Infinitesimal

In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is.

See Limit (mathematics) and Infinitesimal

Integral

In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations.

See Limit (mathematics) and Integral

Inverse limit

In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects.

See Limit (mathematics) and Inverse limit

Karl Weierstrass

Karl Theodor Wilhelm Weierstrass (Weierstraß; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis".

See Limit (mathematics) and Karl Weierstrass

Limit (category theory)

In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits.

See Limit (mathematics) and Limit (category theory)

Limit inferior and limit superior

In 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. Limit (mathematics) and limit inferior and limit superior are limits (mathematics).

See Limit (mathematics) and Limit inferior and limit superior

Limit of a function

Although the function is not defined at zero, as becomes closer and closer to zero, becomes arbitrarily close to 1. Limit (mathematics) and Limit of a function are limits (mathematics).

See Limit (mathematics) and Limit of a function

Limit of a sequence

As the positive integer n becomes larger and larger, the value n\times \sin\left(\tfrac1\right) becomes arbitrarily close to 1. Limit (mathematics) and Limit of a sequence are limits (mathematics).

See Limit (mathematics) and Limit of a sequence

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.

See Limit (mathematics) and Limit set

List of limits

This is a list of limits for common functions such as elementary functions. Limit (mathematics) and list of limits are limits (mathematics).

See Limit (mathematics) and List of limits

Lp space

In mathematics, the spaces are function spaces defined using a natural generalization of the ''p''-norm for finite-dimensional vector spaces.

See Limit (mathematics) and Lp space

Mathematical analysis

Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.

See Limit (mathematics) and Mathematical analysis

Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

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Method of exhaustion

The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape.

See Limit (mathematics) and Method of exhaustion

Metric space

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points.

See Limit (mathematics) and Metric space

Modes of convergence

In mathematics, there are many senses in which a sequence or a series is said to be convergent. Limit (mathematics) and Modes of convergence are convergence (mathematics).

See Limit (mathematics) and Modes of convergence

Modes of convergence (annotated index)

The purpose of this article is to serve as an annotated index of various modes of convergence and their logical relationships. Limit (mathematics) and modes of convergence (annotated index) are convergence (mathematics).

See Limit (mathematics) and Modes of convergence (annotated index)

Modulus of convergence

In real analysis, a branch of mathematics, a modulus of convergence is a function that tells how quickly a convergent sequence converges. Limit (mathematics) and modulus of convergence are real analysis.

See Limit (mathematics) and Modulus of convergence

Natural number

In mathematics, the natural numbers are the numbers 0, 1, 2, 3, etc., possibly excluding 0.

See Limit (mathematics) and Natural number

Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. Limit (mathematics) and neighbourhood (mathematics) are general topology.

See Limit (mathematics) and Neighbourhood (mathematics)

Net (mathematics)

In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a function whose domain is a directed set. Limit (mathematics) and net (mathematics) are general topology.

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Nonstandard analysis

The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers.

See Limit (mathematics) and Nonstandard analysis

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).

See Limit (mathematics) and Numerical analysis

One-sided limit

In calculus, a one-sided limit refers to either one of the two limits of a function f(x) of a real variable x as x approaches a specified point either from the left or from the right. Limit (mathematics) and one-sided limit are limits (mathematics) and real analysis.

See Limit (mathematics) and One-sided limit

Pointwise convergence

In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. Limit (mathematics) and pointwise convergence are convergence (mathematics).

See Limit (mathematics) and Pointwise convergence

Radius of convergence

In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. Limit (mathematics) and radius of convergence are convergence (mathematics).

See Limit (mathematics) and Radius of convergence

Rate of convergence

In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit.

See Limit (mathematics) and Rate of convergence

Ratio test

In mathematics, the ratio test is a test (or "criterion") for the convergence of a series where each term is a real or complex number and is nonzero when is large.

See Limit (mathematics) and Ratio test

Rational number

In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

See Limit (mathematics) and Rational number

Real analysis

In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions.

See Limit (mathematics) and Real analysis

Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.

See Limit (mathematics) and Real number

Real-valued function

In mathematics, a real-valued function is a function whose values are real numbers. Limit (mathematics) and real-valued function are general topology.

See Limit (mathematics) and Real-valued function

Riemann series theorem

In mathematics, the Riemann series theorem, also called the Riemann rearrangement theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, or diverges.

See Limit (mathematics) and Riemann series theorem

Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters.

See Limit (mathematics) and Sequence

Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number, called differentiability class, of continuous derivatives it has over its domain.

See Limit (mathematics) and Smoothness

Sobolev space

In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order.

See Limit (mathematics) and Sobolev space

Squeeze theorem

In calculus, the squeeze theorem (also known as the sandwich theorem, among other names) is a theorem regarding the limit of a function that is bounded between two other functions. Limit (mathematics) and squeeze theorem are limits (mathematics).

See Limit (mathematics) and Squeeze theorem

Standard part function

In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers.

See Limit (mathematics) and Standard part function

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.

See Limit (mathematics) and Subsequence

Topological space

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. Limit (mathematics) and topological space are general topology.

See Limit (mathematics) and Topological space

Topology

Topology (from the Greek words, and) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.

See Limit (mathematics) and Topology

Undecidable problem

In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer.

See Limit (mathematics) and Undecidable problem

Uniform convergence

In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. Limit (mathematics) and uniform convergence are convergence (mathematics).

See Limit (mathematics) and Uniform convergence

Value (mathematics)

In mathematics, value may refer to several, strongly related notions.

See Limit (mathematics) and Value (mathematics)

0.999...

In mathematics, 0.999... (also written as 0., 0., or 0.(9)) denotes the smallest number greater than every number in the sequence.

See Limit (mathematics) and 0.999...

References

[1] https://en.wikipedia.org/wiki/Limit_(mathematics)

Also known as Converge (topology), Convergence (math), Convergence (mathematical), Convergence (maths), Infinite limit, Limit (calculus), Limit (math), Limit (mathematic), Limit (maths), Limit (topology), Limit function, Limit operator, Limit process, Math limit, Mathematical Limit, Mathematical convergence.

, Limit set, List of limits, Lp space, Mathematical analysis, Mathematics, Method of exhaustion, Metric space, Modes of convergence, Modes of convergence (annotated index), Modulus of convergence, Natural number, Neighbourhood (mathematics), Net (mathematics), Nonstandard analysis, Numerical analysis, One-sided limit, Pointwise convergence, Radius of convergence, Rate of convergence, Ratio test, Rational number, Real analysis, Real number, Real-valued function, Riemann series theorem, Sequence, Smoothness, Sobolev space, Squeeze theorem, Standard part function, Subsequence, Topological space, Topology, Undecidable problem, Uniform convergence, Value (mathematics), 0.999....

Limit (mathematics), the Glossary

Table of Contents

See also

Convergence (mathematics)

Limits (mathematics)

References