**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Limit (mathematics)

Summary

In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits 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 and direct limit in .
In formulas, a limit of a function is usually written as
(although a few authors use "Lt" instead of "lim")
and is read as "the limit of f of x as x approaches c equals L". The fact that a function f approaches the limit L as x approaches c is sometimes denoted by a right arrow (→ or ), as in
which reads " of tends to as tends to ".
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."
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 is due to G. H. Hardy, who introduced it in his book A Course of Pure Mathematics in 1908.
Limit of a sequence
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 a1, a2, ... is a sequence of real numbers.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (147)

Related people (32)

Related concepts (34)

Related MOOCs (20)

Related units (12)

Related courses (31)

Convergent series

In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence defines a series S that is denoted The nth partial sum Sn is the sum of the first n terms of the sequence; that is, A series is convergent (or converges) if the sequence of its partial sums tends to a limit; that means that, when adding one after the other in the order given by the indices, one gets partial sums that become closer and closer to a given number.

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. This implies that a series of real numbers is absolutely convergent if and only if it is unconditionally convergent.

Real-valued function

In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called real functions) and real-valued functions of several real variables are the main object of study of calculus and, more generally, real analysis. In particular, many function spaces consist of real-valued functions. Let be the set of all functions from a set X to real numbers .

Analyse I

Le contenu de ce cours correspond à celui du cours d'Analyse I, comme il est enseigné pour les étudiantes et les étudiants de l'EPFL pendant leur premier semestre. Chaque chapitre du cours correspond

Analyse I (partie 1) : Prélude, notions de base, les nombres réels

Concepts de base de l'analyse réelle et introduction aux nombres réels.

Analyse I (partie 2) : Introduction aux nombres complexes

Introduction aux nombres complexes

Related lectures (298)

MATH-101(a): Analysis I

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.

MATH-101(g): Analysis I

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.

MATH-100(a): Advanced analysis I

Nous étudions les concepts fondamentaux de l'analyse, le calcul différentiel et intégral de fonctions réelles d'une variable.

Convergence CriteriaMATH-101(g): Analysis I

Covers the convergence criteria for sequences, including operations on limits and sequences defined by recurrence.

Analysis 1: Fundamentals of Real Analysis

Covers the basics of real analysis, focusing on functions of a real variable.

, , , ,

Time reversal exploits the invariance of electromagnetic wave propagation in reciprocal and lossless media to localize radiating sources. Time-reversed measurements are back-propagated in a simulated domain and converge to the unknown source location. The ...

2024Nikolaos Stergiopulos, Sokratis Anagnostopoulos

Driven by the need for more efficient and seamless integration of physical models and data, physics -informed neural networks (PINNs) have seen a surge of interest in recent years. However, ensuring the reliability of their convergence and accuracy remains ...

, , ,

The construction industry's carbon footprint may be reduced by switching to the usage of natural stones as building materials. Construction of dry-joint stone masonry structures is difficult because of the irregular shapes and the nonuniform sizes of the s ...

2023