Limit (mathematics)
Encyclopedia
In mathematics
, the concept of a "limit" is used to describe the value that a function
or sequence
"approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence
of previously defined points within a complete metric space. Limits are essential to calculus
(and mathematical analysis
in general) and are used to define continuity
, derivative
s, and integral
s.
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 limit
and direct limit
in category theory
.
In formulas, limit is usually abbreviated as lim as in lim(an) = a or represented by the right arrow (→) as in an → a.
and is a real number
. The expression
means that can be made to be as close to as desired by making sufficiently close to . In that case, it can be stated that "the limit of of , as approaches , is ". Note that this statement can be true even if . Indeed, the function need not even be defined at .
For example, if
then f(1) is not defined (see division by zero
), yet as moved arbitrarily close to 1, correspondingly approaches 2:
Thus, can be made arbitrarily close to the limit of 2 just by making sufficiently close to 1.
Augustin-Louis Cauchy in 1821, followed by Karl Weierstrass
, formalized the definition of the limit of a function into what became known as the (ε, δ)-definition of limit in the 19th century. The definition uses (the lowercase Greek letter epsilon) to represent a small positive number, so that " becomes arbitrarily close to " means that lies in the interval , which can also be written using absolute value as . The statement " approaches " then indicates that a positive number (the lowercase greek letter delta) exists within either or , which can be expressed with . The first inequality means that the distance between and is more than 0 and that , while the second indicates that is within distance of .
In addition to limits at finite values, functions can also have limits at infinity. For example, consider
As becomes extremely large, the value of approaches 2, and the value of can be made as close to 2 as one could wish just by picking sufficiently large. In this case, the limit of as approaches infinity is 2. In mathematical notation,
Formally, suppose , , ... is a sequence
of real number
s. It can be stated that the real number is the limit of this sequence, namely:
to mean
Intuitively, this means that eventually all elements of the sequence get as close as needed to the limit, since the absolute value
|n − | is the distance between n and . Not every sequence has a limit; if it does, it is called convergent, and if it does not, it is divergent. One can show that a convergent sequence has only one limit.
The limit of a sequence and the limit of a function are closely related. On one hand, the limit of a sequence is simply the limit at infinity of a function defined on natural number
s. On the other hand, a limit of a function f at , if it exists, is the same as the limit of the sequence f(an) where an is any arbitrary sequence whose limit is x, and where an is never equal to x. Note that one such sequence would be .
Suppose
as 
goes from 
to 
is a sequence that converges to a fixed point 
, with 
for all 
. If positive constants 
and 
exist with
then
as 
goes from 
to 
converges to 
of order 
, with asymptotic error constant 
Given a function
with a fixed point 
, there is a nice checklist for checking the convergence of p.
s by introducing topological nets and defining their limits.
An alternative is the concept of limit for filters
on topological spaces.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the concept of a "limit" is used to describe the value that a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
or sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
"approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence
Cauchy sequence
In mathematics, a Cauchy sequence , named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses...
of previously defined points within a complete metric space. Limits are essential to calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
(and mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
in general) and are used to define continuity
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
, derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
s, and integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
s.
The concept of a limit of a sequence
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
is further generalized to the concept of a limit of a topological net, and is closely related to limit
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 and inverse limits....
and direct limit
Direct limit
In mathematics, a direct limit is a colimit of a "directed family of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category.- Algebraic objects :In this section objects are understood to be...
in category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
.
In formulas, limit is usually abbreviated as lim as in lim(an) = a or represented by the right arrow (→) as in an → a.
Limit of a function
Suppose is a real-valued functionReal-valued function
In mathematics, a real-valued function is a function that associates to every element of the domain a real number in the image....
and is a real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
. The expression
means that can be made to be as close to as desired by making sufficiently close to . In that case, it can be stated that "the limit of of , as approaches , is ". Note that this statement can be true even if . Indeed, the function need not even be defined at .
For example, if
then f(1) is not defined (see division by zero
Division by zero
In mathematics, division by zero is division where the divisor is zero. Such a division can be formally expressed as a / 0 where a is the dividend . Whether this expression can be assigned a well-defined value depends upon the mathematical setting...
), yet as moved arbitrarily close to 1, correspondingly approaches 2:
| f(0.9) | f(0.99) | f(0.999) | f(1.0) | f(1.001) | f(1.01) | f(1.1) |
| 1.900 | 1.990 | 1.999 | ⇒ undef ⇐ | 2.001 | 2.010 | 2.100 |
Thus, can be made arbitrarily close to the limit of 2 just by making sufficiently close to 1.
Augustin-Louis Cauchy in 1821, followed by Karl Weierstrass
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....
, formalized the definition of the limit of a function into what became known as the (ε, δ)-definition of limit in the 19th century. The definition uses (the lowercase Greek letter epsilon) to represent a small positive number, so that " becomes arbitrarily close to " means that lies in the interval , which can also be written using absolute value as . The statement " approaches " then indicates that a positive number (the lowercase greek letter delta) exists within either or , which can be expressed with . The first inequality means that the distance between and is more than 0 and that , while the second indicates that is within distance of .
In addition to limits at finite values, functions can also have limits at infinity. For example, consider
- f(100) = 1.9900
- f(1000) = 1.9990
- f(10000) = 1.99990
As becomes extremely large, the value of approaches 2, and the value of can be made as close to 2 as one could wish just by picking sufficiently large. In this case, the limit of as approaches infinity is 2. In mathematical notation,
Limit of a sequence
Consider the following sequence: 1.79, 1.799, 1.7999,... It can be observed that the numbers are "approaching" 1.8, the limit of the sequence.Formally, suppose , , ... is a sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s. It can be stated that the real number is the limit of this sequence, namely:
to mean
- For every real numberReal numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
ε > 0, there exists a natural numberNatural numberIn mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
n0 such that for all n > n0, |n − | < ε.
Intuitively, this means that eventually all elements of the sequence get as close as needed to the limit, since the absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
|n − | is the distance between n and . Not every sequence has a limit; if it does, it is called convergent, and if it does not, it is divergent. One can show that a convergent sequence has only one limit.
The limit of a sequence and the limit of a function are closely related. On one hand, the limit of a sequence is simply the limit at infinity of a function defined on natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s. On the other hand, a limit of a function f at , if it exists, is the same as the limit of the sequence f(an) where an is any arbitrary sequence whose limit is x, and where an is never equal to x. Note that one such sequence would be .
Convergence and fixed point
A formal definition of convergence can be stated as follows.Suppose
as goes from to is a sequence that converges to a fixed point , with for all . If positive constants and exist with
then
as goes from to converges to of order , with asymptotic error constant Given a function
with a fixed point , there is a nice checklist for checking the convergence of p.- 1) First check that p is indeed a fixed point:
- 2) Check for linear convergence. Start by finding
. If....
then there is linear convergence series diverges then there is at least linear convergence and maybe something better, the expression should be checked for quadratic convergence - 3) If it is found that there is something better than linear the expression should be checked for quadratic convergence. Start by finding
If....
then there is quadratic convergence provided that 
is continuousthen there is something even better than quadratic convergence does not exist then there is convergence that is better than linear but still not quadratic
Topological net
All of the above notions of limit can be unified and generalized to arbitrary topological spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s by introducing topological nets and defining their limits.
An alternative is the concept of limit for filters
Filter (mathematics)
In mathematics, a filter is a special subset of a partially ordered set. A frequently used special case is the situation that the ordered set under consideration is just the power set of some set, ordered by set inclusion. Filters appear in order and lattice theory, but can also be found in...
on topological spaces.
See also
- Limit of a sequenceLimit of a sequenceThe limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
- Rate of convergenceRate of convergenceIn numerical analysis, the speed at which a convergent sequence approaches its limit is called the rate of convergence. Although strictly speaking, a limit does not give information about any finite first part of the sequence, this concept is of practical importance if we deal with a sequence of...
: the rate at which a convergent sequence approaches its limit
- Rate of convergence
- Limit of a functionLimit of a functionIn mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....
- One-sided limitOne-sided limitIn calculus, a one-sided limit is either of the two limits of a function f of a real variable x as x approaches a specified point either from below or from above...
: either of the two limits of functions of a real variable x, as x approaches a point from above or below - List of limits: list of limits for common functions
- Squeeze theoremSqueeze theoremIn calculus, the squeeze theorem is a theorem regarding the limit of a function....
: finds a limit of a function via comparison with two other functions
- One-sided limit
- Limit in category theoryLimit (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 and inverse limits....
- Direct limitDirect limitIn mathematics, a direct limit is a colimit of a "directed family of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category.- Algebraic objects :In this section objects are understood to be...
- Direct limit
- Asymptotic analysisAsymptotic analysisIn mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The methodology has applications across science. Examples are...
: a method of describing limiting behavior- Big O notationBig O notationIn mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann-Landau notation, or...
: used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity
- Big O notation