Difference between revisions of "Limit"
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− | One writes $\lim_{n\to\infty} x_n = \infty$ ($\lim_{n\to \infty} x_n = +\infty$ or $-\infty$), $x_n\in \mathbb R$, $n=1,2,\dots,$ if for each $\epsilon>0$ there is an index $N$ such that for all indices $n>N$ the membership $x_n\in U(\infty,\epsilon)$ ($x_n\in U(+\infty,\epsilon$ or $x_n\in U(-\infty,\epsilon)$) is satisfied. The infinite limit of a sequence of complex numbers is defined similarly. | + | One writes $\lim_{n\to\infty} x_n = \infty$ ($\lim_{n\to \infty} x_n = +\infty$ or $-\infty$), $x_n\in \mathbb R$, $n=1,2,\dots,$ if for each $\epsilon>0$ there is an index $N$ such that for all indices $n>N$ the membership $x_n\in U(\infty,\epsilon)$ ($x_n\in U(+\infty,\epsilon)$ or $x_n\in U(-\infty,\epsilon)$) is satisfied. The infinite limit of a sequence of complex numbers is defined similarly. |
Every bounded sequence of numbers contains a convergent subsequence (cf. [[Bolzano–Weierstrass theorem|Bolzano–Weierstrass theorem]]). Every unbounded sequence contains an infinitely-large sequence. | Every bounded sequence of numbers contains a convergent subsequence (cf. [[Bolzano–Weierstrass theorem|Bolzano–Weierstrass theorem]]). Every unbounded sequence contains an infinitely-large sequence. |
Revision as of 20:42, 29 November 2013
One of the fundamental concepts in mathematics, meaning that a variable depending on another variable arbitrary closely approaches some constant as the latter variable changes in a definite manner. In the definition of a limit, the concept of nearness of the objects under consideration plays a fundamental role: only after a definition of nearness does a limit acquire an exact meaning. The following fundamental concepts of mathematical analysis are connected with that of a limit: continuity, derivative, differential, integral. One of the simplest cases of a limit is the limit of a sequence.
The limit of a sequence.
Let $X$ be a topological space. A sequence of points $x_n$, $n=1,2,\dots,$ of $X$ is said to converge to a point $x_0\in X$ or, which is the same, the point $x_0$ is said to be a limit of the given sequence if for each neighbourhood $U$ of $x_0$ there is a natural number $N$ such that for all $n>N$ the membership $x_n\in U$ is satisfied. In this case one writes $$ \lim_{n\to \infty} x_n = x_0. $$ In the case when $X$ is a Hausdorff space, the limit of a sequence $x_n\in X$, $n=1,2,\dots,$ is unique, provided that it exists. For a metric space $X$, a point $x_0$ is the limit of a sequence $\{x_n\}$ if and only if for each $\epsilon >0$ there is a natural number $N$ such that for all indices $n>N$ the inequality $\rho(x_n,x_0)<\epsilon$ is satisfied, where $\rho(x,x_0)$ is the distance between $x_n$ and $x_0$. If a sequence of points of a metric space is convergent, then it is bounded. A sequence of points of a complete metric space is convergent if and only if it is a fundamental sequence. In particular, this is true for sequences of numbers, for which the concept of a limit of a sequence historically arose first. For such sequences the following formulas hold: $$ \lim_{n\to \infty} c = c, \quad \lim_{n\to \infty} cx_n = c \lim_{n\to \infty} x_n, $$ where $c$ is any given number; $$ \lim_{n\to \infty} (x_n+y_n) = \lim_{n\to \infty} x_n + \lim_{n\to\infty}y_n;\\ \lim_{n\to \infty} x_n y_n = \lim_{n\to \infty} x_n \lim_{n\to \infty} y_n; $$ and if $\lim_{n\to \infty} y_n \neq 0$, then $$ \lim_{n\to\infty}\frac{x_n}{y_n} = \frac{\lim_{n\to \infty} x_n}{\lim_{n\to\infty} y_n}. $$ These properties of sequences of numbers can be carried over to limits of sequences in more general structures, for example, the property of the limit of a sum — to sequences of points in linear topological spaces, the property of the limit of a product — to sequences of points in a topological group, etc.
If two real sequences $x_n\in\mathbb R$ and $y_n\in \mathbb R$ are convergent and if $x_n \le y_n$, $n=1,2,\dots,$ then $$\lim_{n\to \infty} x_n \le \lim_{n\to \infty} y_n, $$ i.e. non-strict inequalities are preserved under limit transition. If $$\lim_{n\to \infty} x_n = \lim_{n\to \infty} y_n = \alpha$$and if $x_n\le z_n \le y_n$, then the sequence $z_n$, $n=1,2,\dots,$ converges to the same limit: $\lim_{n\to \infty} z_n = \alpha$. These properties can be generalized to limits of sequences of points in ordered sets.
Every increasing (decreasing) sequence of real numbers $x_n$, i.e. such that $x_n \le x_{n+1}$ ($x_n \ge x_{n+1}$), $n=1,2,\dots,$ that is bounded from above (below) is convergent and its limit is the supremum (infimum) of the set of its members. For example, if $\alpha>0$, $k$ is a natural number and $\alpha_n$ is an approximate value of the root $\alpha^{1/k}$ calculated up to $n$ decimal places after the decimal point, then the $\alpha_n$, $n=1,2,\dots,$ form an increasing sequence and $\lim_{n\to\infty}\alpha_n = \alpha^{1/k}$. Another example of an increasing sequence which is bounded from above is the sequence of perimeters of regular polygons with $n$ sides, $n=3,4,\dots,$ inscribed in some circle; this sequence converges to the length of the circle.
In the theory of sequences of numbers a fundamental role is played by infinitesimal sequences or null sequences, i.e. those sequences which converge to zero. The general concept of a sequence of numbers can be reduced to that of an infinitesimal sequence in the sense that a sequence of numbers converges to a given number if and only if the differences between the terms of the sequence and the given number form an infinitesimal sequence.
The concept of an infinitely-large sequence of numbers is also useful. These are the sequences with as limits one of the infinities $+\infty$, $-\infty$, or the infinity $\infty$ without a sign. For the definition of infinite limits, the concept of $\epsilon$-neighbourhoods, $\epsilon>0$, of the symbols $+\infty$, $-\infty$ or $\infty$ in the set $\mathbb R$ of real numbers is introduced by the formulas $$ U(+\infty,\epsilon) = \left\{ x \in \mathbb R \colon x > \frac{1}{\epsilon}\right\},\\ U(-\infty,\epsilon) = \left\{ x \in \mathbb R \colon x < -\frac{1}{\epsilon}\right\},\\ U(\infty,\epsilon) = \left\{ x \in \mathbb R \colon \lvert x \rvert > \frac{1}{\epsilon}\right\},\\ $$ and the concept of the $\epsilon$-neighbourhood of $\infty$ in the set $\mathbb C$ of complex numbers is introduced by the formula $$ U(\infty,\epsilon) = \left\{ z \in \mathbb C\colon \lvert z \rvert > \frac 1 \epsilon\right\}. $$
One writes $\lim_{n\to\infty} x_n = \infty$ ($\lim_{n\to \infty} x_n = +\infty$ or $-\infty$), $x_n\in \mathbb R$, $n=1,2,\dots,$ if for each $\epsilon>0$ there is an index $N$ such that for all indices $n>N$ the membership $x_n\in U(\infty,\epsilon)$ ($x_n\in U(+\infty,\epsilon)$ or $x_n\in U(-\infty,\epsilon)$) is satisfied. The infinite limit of a sequence of complex numbers is defined similarly.
Every bounded sequence of numbers contains a convergent subsequence (cf. Bolzano–Weierstrass theorem). Every unbounded sequence contains an infinitely-large sequence.
A (finite or infinite) limit of a subsequence of a given sequence is called a subsequential limit of the latter. In the set of subsequential limits of any sequence of real numbers there is always a largest one and a smallest one (finite or infinite). The largest (smallest) subsequential limit of a sequence is called its upper (lower) limit. A sequence has a finite or infinite limit if and only if its upper limit coincides with its lower limit, and then their common value is the limit of the sequence.
Other concepts of a limit, for example, the limit of a function and of Riemann sums, can be expressed in terms of the limit of a sequence. The definition of the limit of a sequence can be generalized to directed (partially ordered) sets.
The limit of a function (mapping).
Let and
be topological spaces,
,
an accumulation point (or a cluster point) of
, and let
be a mapping from
into
. A point
is called a limit of the mapping
at
(or, as one says, as
approaches
), in symbols,
![]() |
if for any neighbourhood of
in
there is a neighbourhood
of
in
, such that for any point
the image
belongs to
:
. In other words, if
. If
is a Hausdorff space, then the mapping
can have only one limit at a given point
.
In the case where and
is an accumulation point of
, a limit of the restriction
of
to
is called a limit of
on
, in symbols,
![]() |
If ,
,
is an accumulation point of
, and
exists, then the limit of
at
also exists on
and
![]() |
If ,
and the limits
![]() |
exist, then the limit
![]() |
also exists.
In considering the limit of a mapping (function) ,
, as
, it may happen that
or, on the other hand,
. The case
is of special interest, because it leads to the concept of a continuous function: If
,
is a Hausdorff space and
, then for the mapping
to be continuous at
it is necessary and sufficient that
![]() |
If is an isolated point of
, then the limit
![]() |
always exists, for any mapping , i.e. any mapping
is continuous at all isolated points of its domain. Therefore, the concept of a limit of a mapping, in particular, of continuity, is non-trivial only for limit points of the set being mapped (cf. Limit point of a set). In the classical case of the limit of a function
it is usually assumed that
, i.e.
does not belong to the set on which the limit is taken.
If the space satisfies the first axiom of countability at the point
and the space
is Hausdorff, then for the existence of the limit
of a mapping
,
, it is necessary and sufficient that for any sequence
,
such that
, the limit
exists. If this condition holds, the limit
does not depend on the choice of the sequence
, and the common value of these limits is the limit of
at
.
The limit of a sequence of points in a topological space
is a special case of the limit of a mapping (function): in this case
, the set of natural numbers with the discrete topology,
,
, and a neighbourhood of
in
is any subset
of the form
, where
is a natural number.
The concept of the limit of a multiple sequence, i.e. of a sequence the members of which are indexed by integral multi-indices, is also a special case of the limit of a mapping.
An intrinsic criterion for the existence of the limit of a mapping at a given point
(called the Cauchy criterion) in the case where the space
is first countable at
and
is a complete metric space, is that the limit
exists if and only if for each
there is a neighbourhood
of
in
such that for all points
and
satisfying the condition
, the inequality
holds. In particular, this criterion is valid if
is the set of real or complex numbers.
Some properties of the limit.
If is a metric space,
,
, and the limit
exists, then there is a neighbourhood
of
such that the image of the intersection
of the set
being mapped and the neighbourhood
under the mapping
is a bounded subset of
.
If a function ,
, where
is the set of real numbers, has a finite non-zero limit at a point
, then there exist a neighbourhood
of
and a number
such that for all points
the inequalities
![]() |
![]() |
are satisfied.
If is a topological group (in particular, an Abelian group with group operation written additively),
,
, and
, then the limit
exists and equals
if and only if the function
has a limit at
which is equal to the identity element of
(respectively, the function
has a limit at
which is equal to zero — such functions are called infinitesimal functions).
If is a linear topological space over a field
,
, and
, then the limit of a linear combination of
and
at
is equal to the same linear combination of their limits at the same point:
![]() |
![]() |
If is the set of real or complex numbers,
(such functions are called numerical) and
, then
![]() |
if , then
![]() |
and in this case, by the limit is meant the limit of the restriction of
to the intersection of the set
being mapped and some neighbourhood of
, such that on this intersection the quotient
is defined. If
,
, and the limits
and
exist, then
![]() |
If and
are obtained from the set
of real numbers by completing it either with an infinity without a sign
or by two signed infinities
and
, and
,
, and
, then the definition of the limit of a function as defined above is the classical definition of a finite or infinite limit of a real-valued function of one real variable. Similarly, if the spaces
and
are obtained by completing the set
of complex numbers by the infinity
, then the definition of a (finite or infinite) limit of a function of a complex variable is obtained. On the other hand, if the space
is obtained by completing
(
),
, by the infinity
, then the definition of a finite or infinite limit of a function of several variables as the argument approaches a finite point or infinity is obtained.
For functions defined on subsets of the real line (or, more generally, on ordered sets), the concept of a one-sided limit is defined. Examples of functions having at least one one-sided limit at all limit points of their domain are the real-valued monotone functions: If a function is monotone (cf. Monotone function) on a set
of the real line and the point
is a limit point of
, then it is a limit point of at least one of the sets
or
. If
is a limit point of
, then
has a limit from the left at
, i.e. on
. On the other hand, if
is a limit point of
, then
has a limit from the right at
i.e. on
. If, in addition, for example,
is increasing and bounded from above,
and
is a limit point of
, then the limit
is finite.
A fundamental general method for finding the limit of a function is the determination of the principal parts of the function in a neighbourhood of the given point, which is often done by means of the Taylor formula. For the calculation of a limit, the l'Hospital rule is often useful.
In spite of the great generality of the concept of a limit of a mapping, it does not include all existing concepts of a limit appearing in contemporary mathematics. For example, the concept of the limit of Riemann sums (cf. Integral sum) is not included in the concept of the limit of a mapping (function). A sufficiently general concept of a limit, including all fundamental cases in a certain sense, is the concept of the limit of a mapping with respect to a filter.
The limit of a filter.
Let be a topological space,
, let
be a base for its topology, and let
be a filter on
(i.e. a non-empty family
of non-empty subsets of
such that for any
there is an
such that
and such that every subset of
containing an
belongs to
). A point
is called a limit of the filter
, or its limit point, if
is stronger than the filter
consisting of a local base for the topology at
, i.e. for any
there is an
such that
.
Let be the set of natural numbers with the discrete topology. The filter on
consisting of the complements of all finite subsets of
is called the natural filter on
and is denoted by
. It does not have a limit in
. The same filter on the set
, in which the local base
consists of the sets
and
consists of the singleton
for
, has
as its limit. Uniqueness of the limit of a filter on a topological space is connected with being able to separate points of the space; in order that every filter on a topological space has at most one limit it is necessary and sufficient that the space be a Hausdorff space.
Let be a set,
a topological space,
a mapping from
into
, and
a filter on
. A point
is called the limit of the mapping
with respect to the filter
, in symbols,
![]() |
if the filter consisting of all sets
,
, has
as its limit in
.
If is the set of natural numbers,
is a mapping from
into a topological space
,
,
, and
is the natural filter, then the limit of
with respect to
in
coincides with the usual limit of the sequence
in
.
If in , the filter
on
is the product of two natural filters
, i.e. it consists of all sets of the form
, where
, and if
is a mapping from
into a topological space
,
,
, then the limit of
with respect to
in
coincides with the usual limit of the double sequence
in
.
Let the elements of a set , in turn, be the sets
consisting of a partition
of some interval
,
, and points
,
i.e.
![]() |
Let (for any
) be the subset of
consisting of all elements
for which the mesh of the partition
appearing in
is smaller than
, i.e.
![]() |
The system is a filter. Every real-valued function
defined on
induces a mapping
of
into
by the formula
![]() |
![]() |
Therefore, is the Riemann sum of
corresponding to
.
The limit of in
with respect to
coincides with the usual limit of the Riemann sums of
as the mesh of the partition converges to zero. This coincidence holds in the sense that both limits simultaneously exist or not, and if they do so, then they are equal and coincide with the Riemann integral of
over
.
The limit of a mapping of topological spaces with respect to a filter.
Let and
be topological spaces,
, let
be a filter on
, and let
be a mapping from
into
. A point
is called the limit of the mapping
at
with respect to the filter
if
is the limit of
and
is the limit of
; notation:
![]() |
If is a neighbourhood base at
,
and the filter consists of all "punctured" neighbourhoods
of
,
, then the limit
coincides with the usual limit
of
at
, i.e. it generalizes the classical definition of the limit of a mapping as formulated in terms of neighbourhoods. An immediate generalization of the concept of the limit of a sequence is the limit of a directed set in a topological space, i.e. a partially ordered set in which any two elements have a common successor. The concept of a limit of a mapping from one topological space into another can be formulated in terms of limits with respect to directed sets (cf. Generalized sequence; Convergence, types of).
The limit of a sequence of sets.
The topological limit. Let ,
be subsets of a topological space
. The upper topological limit
of the sequence
is, by definition, the set of those points
every neighbourhood of which intersects infinitely many sets
. The lower topological limit
is the set of those points every neighbourhood of which contains points of almost-all sets
. It is obvious that
. If
, then the sequence
is called convergent and the set
its topological limit; one writes
. The upper and lower topological limits of a sequence are closed sets.
The set-theoretical limit. There is a concept of the limit of a sequence of sets not involving topology. A sequence ,
of sets is called convergent if there is a set
, called its limit and denoted by
![]() |
such that every element of belongs to all sets
from some index onwards and such that every point of the union of all
not belonging to
is contained in only finitely many sets
. The set
is the limit of the sequence
if and only if the upper and lower limits of the sequence coincide and are equal to
.
References
[1] | V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian) |
[2] | V.A. Il'in, V.A. Sadovnichii, B.Kh. Sendov, "Mathematical analysis" , Moscow (1979) (In Russian) |
[3] | L.D. Kudryavtsev, "A course in mathematical analysis" , 1–3 , Moscow (1988–1989) (In Russian) |
[4] | S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1977) (Translated from Russian) |
[5] | N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French) |
[6] | M. Zamansky, "Introduction à l'algèbre et l'analyse modernes" , Dunod (1958) |
[7] | A. Krazer, W. Franz, "Transzendente Funktionen" , Akademie Verlag (1960) |
[8] | F. Hausdorff, "Grundzüge der Mengenlehre" , Leipzig (1914) (Reprinted (incomplete) English translation: Set theory, Chelsea (1978)) |
Comments
For the general theory of topological spaces one needs to study more general convergence concepts than that of sequences of points. One needs to consider limits of sets of points indexed by a directed partially ordered set (cf. Directed set). Such sets are called nets, and the convergence of nets is sometimes known as Moore–Smith convergence.
The basic (inductive) idea of "limit" is that of some object being approximated arbitrarily closely by a sequence of other objects. There are also various algebraic and categorical implementations of this idea.
Let be a category and
a partially ordered set. A diagram
in
indexed by
consists of an object
for each
and a morphism
for each
in
, such that
,
if
.
The projective limit (or inverse limit) of
consists of an object
together with morphisms
such that
if
and such that if, moreover,
and
are such that
for all
, then there is a unique morphism
such that
. Notation:
.
Dually, the inductive limit (or directed limit) of
consists of an object
together with morphisms
such that
for all
and such that if, moreover,
and
are such that
for all
, then there is a unique morphism
such that
. Notation:
.
These are very general notions and include direct products and direct sums (when every pair of points in is incomparable). In case
is directed (cf. Directed set), the idea of better and better approximations again reemerges; cf. e.g. Topological vector space.
More generally one also considers projective and inductive limits of diagrams in categories (cf. Diagram).
Still another limit concept is that of the limit of a spectral sequence: If the spectral sequence converges to, say,
, then
is also called the limit of
.
Still other limit ideas are embodied, for example, by the concepts limit cycle; limit elements (cf. the supplementary material section of Vol. 10); limit point of a set; accumulation point; cluster point; etc.
References
[a1] | R. Courant, "Differential and integral calculus" , 1–2 , Blackie (1948) (Translated from German) |
[a2] | K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981) |
[a3] | J.L. Kelley, "General topology" , v. Nostrand (1955) pp. 125; 127 |
[a4] | B. Mitchell, "Theory of categories" , Acad. Press (1965) pp. 7 |
[a5] | J. Adámek, "Theory of mathematical structures" , Reidel (1983) |
Limit. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Limit&oldid=30820