In mathematics, a small positive infinitesimal quantity, usually denoted or , whose limit is usually taken as .The late mathematician P. Erdős also used the term "epsilons" to refer to children (Hoffman 1998, p. 4).
The term asymptotic means approaching a value or curve arbitrarily closely (i.e., as some sort of limit is taken). A line or curve that is asymptotic to given curve is called the asymptote of .Hardy and Wright (1979, p. 7) use the symbol to denote that one quantity is asymptotic to another. If , then Hardy and Wright say that and are of the same order of magnitude.
A function has the normal order if is approximately for almost all values of . More precisely, iffor every positive and almost all values of , then the normal order of is .
Let the least term of a sequence be a term which is smaller than all but a finite number of the terms which are equal to . Then is called the lower limit of the sequence.A lower limit of a seriesis said to exist if, for every , for infinitely many values of and if no number less than has this property.
Let be a nonempty set of real numbers that has a lower bound. A number is the called the greatest lower bound (or the infimum, denoted ) for iff it satisfies the following properties: 1. for all . 2. For all real numbers , if is a lower bound for , then .
Let the greatest term of a sequence be a term which is greater than all but a finite number of the terms which are equal to . Then is called the upper limit of the sequence.An upper limit of a seriesis said to exist if, for every , for infinitely many values of and if no number larger than has this property.
A number such that for all , there exists a member of the set different from such that .The topological definition of limit point of is that is a point such that every open set around it contains at least one point of different from .
The supremum is the least upper bound of a set , defined as a quantity such that no member of the set exceeds , but if is any positive quantity, however small, there is a member that exceeds (Jeffreys and Jeffreys 1988). When it exists (which is not required by this definition, e.g., does not exist), it is denoted (or sometimes simply for short). The supremum is implemented in the Wolfram Language as MaxValue[f, constr, vars].More formally, the supremum for a (nonempty) subset of the affinely extended real numbers is the smallest value such that for all we have . Using this definition, always exists and, in particular, .Whenever a supremum exists, its value is unique. On the real line, the supremum of a set is the same as the supremum of its set closure.Consider the real numbers with their usual order. Then for any set , the supremum exists (in ) if and only if is bounded from above and nonempty...
Let be a nonempty set of real numbers that has an upper bound. Then a number is called the least upper bound (or the supremum, denoted ) for iff it satisfies the following properties: 1. for all . 2. For all real numbers , if is an upper bound for , then .
Let lim stand for the limit , , , , or , and suppose that lim and lim are both zero or are both . If(1)has a finite value or if the limit is , then(2)Historically, this result first appeared in l'Hospital's 1696 treatise, which was the first textbook on differential calculus. Within the book, l'Hospital thanks the Bernoulli brothers for their assistance and their discoveries. An earlier letter by John Bernoulli gives both the rule and its proof, so it seems likely that Bernoulli discovered the rule (Larson et al. 1999, p. 524).Note that l'Hospital's name is commonly seen spelled both "l'Hospital" (e.g., Maurer 1981, p. 426; Arfken 1985, p. 310) and "l'Hôpital" (e.g., Maurer 1981, p. 426; Gray 1997, p. 529), the two being equivalent in French spelling.L'Hospital's rule occasionally fails to yield useful results, as in the case of the function , illustrated above. Repeatedly applying..
Let for all in some open interval containing . Ifthen .
A theorem about (or providing an equivalent definition of) compact sets, originally due to Georg Cantor. Given a decreasing sequence of bounded nonempty closed setsin the real numbers, then Cantor's intersection theorem states that there must exist a point in their intersection, for all . For example, . It is also true in higher dimensions of Euclidean space.Note that the hypotheses stated above are crucial. The infinite intersection of open intervals may be empty, for instance . Also, the infinite intersection of unbounded closed sets may be empty, e.g., .Cantor's intersection theorem is closely related to the Heine-Borel theorem and Bolzano-Weierstrass theorem, each of which can be easily derived from either of the other two. It can be used to show that the Cantor set is nonempty.
The infimum is the greatest lower bound of a set , defined as a quantity such that no member of the set is less than , but if is any positive quantity, however small, there is always one member that is less than (Jeffreys and Jeffreys 1988). When it exists (which is not required by this definition, e.g., does not exist), the infimum is denoted or . The infimum is implemented in the Wolfram Language as MinValue[f, constr, vars].Consider the real numbers with their usual order. Then for any set , the infimum exists (in ) if and only if is bounded from below and nonempty.More formally, the infimum for a (nonempty) subset of the affinely extended real numbers is the largest value such that for all we have . Using this definition, always exists and, in particular, .Whenever an infimum exists, its value is unique...
The term limit comes about relative to a number of topics from several different branches of mathematics.A sequence of elements in a topological space is said to have limit provided that for each neighborhood of , there exists a natural number so that for all . This very general definition can be specialized in the event that is a metric space, whence one says that a sequence in has limit if for all , there exists a natural number so that(1)for all . In many commonly-encountered scenarios, limits are unique, whereby one says that is the limit of and writes(2)On the other hand, a sequence of elements from an metric space may have several - even infinitely many - different limits provided that is equipped with a topology which fails to be T2. One reads the expression in (1) as "the limit as approaches infinity of is ."The topological notion of convergence can be rewritten to accommodate a wider array of topological spaces by utilizing the language..
The Cauchy principal value of a finite integral of a function about a point with is given by(Henrici 1988, p. 261; Whittaker and Watson 1990, p. 117; Bronshtein and Semendyayev 1997, p. 283). Similarly, the Cauchy principal value of a doubly infinite integral of a function is defined byThe Cauchy principal value is also known as the principal value integral (Henrici 1988, p. 261), finite part (Vladimirov 1971), or partie finie (Vladimirov 1971).The Cauchy principal value of an integral having no nonsimple poles can be computed in the Wolfram Language using Integrate[f, x, a, b, PrincipalValue -> True]. Cauchy principal values of functions with possibly nonsimple poles can be computed numerically using the "CauchyPrincipalValue" method in NIntegrate.Cauchy principal values are important in the theory of generalized functions, where they allow extension of results to .Cauchy principal values..