limit (mathematics)

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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, a limit is the value that a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
(or
sequence In , a sequence is an enumerated collection of in which repetitions are allowed and matters. Like a , it contains (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unl ...

) approaches as the input (or index) approaches some
value Value or values may refer to: * Value (ethics) it may be described as treating actions themselves as abstract objects, putting value to them ** Values (Western philosophy) expands the notion of value beyond that of ethics, but limited to Western s ...
. Limits are essential to
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...

and
mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical ...
, and are used to define continuity,
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its Argument of a function, argument (input value). Derivatives are a fundament ...

s, and
integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s. The concept of a
limit of a sequence As the positive integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known ...
is further generalized to the concept of a limit of a topological net, and is closely related to
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
and
direct limit In , 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. These objects may be , , or in general objects from any . The way they are put together is specifi ...
in
category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled directed edges are cal ...
. In formulas, a limit of a function is usually written as :$\lim_ f\left(x\right) = L,$ ::or :, and is read as "the limit of of as approaches equals ". The fact that a function approaches the limit as approaches is sometimes denoted by a right arrow (→ or $\rightarrow$), as in :$f\left(x\right) \to L \text x \to c,$ which reads "$f$ of $x$ tends to $L$ as $x$ tends to $c$".

Limit of a function

Suppose is a
real-valued function Mass measured in grams is a function from this collection of weight to positive number">positive Positive is a property of Positivity (disambiguation), positivity and may refer to: Mathematics and science * Converging lens or positive lens, i ...
and is a
real number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. Intuitively speaking, the expression :$\lim_f\left(x\right) = L$ means that can be made to be as close to as desired, by making sufficiently close to . In that case, the above equation can be read as "the limit of of , as approaches , is ".
Augustin-Louis Cauchy Baron Baron is a rank of nobility or title of honour, often hereditary, in various European countries, either current or historical. The female equivalent is baroness. Typically, the title denotes an aristocrat who ranks higher than a lord ...

in 1821, followed by
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematics, mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university withou ...

, formalized the definition of the limit of a function which became known as the (ε, δ)-definition of limit. The definition uses (the lowercase Greek letter ''epsilon'') to represent any small positive number, so that " becomes arbitrarily close to " means that eventually lies in the interval , which can also be written using the
absolute value of the absolute value function for real numbers In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calc ...

as . The phrase "as approaches " then indicates that we refer to values of , whose distance from is less than some positive number (the lowercase Greek letter ''delta'')—that is, values of within either or , which can be expressed with . The first inequality means that , while the second indicates that is within distance of . The above definition of a limit is true even if . Indeed, the function need not even be defined at . For example, if :$f\left(x\right) = \frac$ then is not defined (see
Indeterminate formIn calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. The ...
), yet as moves arbitrarily close to 1, correspondingly approaches 2: Thus, can be made arbitrarily close to the limit of 2—just by making sufficiently close to . In other words, $\lim_ \frac = 2.$ This can also be calculated algebraically, as $\frac = \frac = x+1$ for all real numbers . Now, since is continuous in at 1, we can now plug in 1 for , leading to the equation $\lim_ \frac = 1+1 = 2.$ In addition to limits at finite values, functions can also have limits at infinity. For example, consider the function $f(x) = \frac$ where: * * * As becomes extremely large, the value of approaches , and the value of can be made as close to as one could wish—by making sufficiently large. So in this case, the limit of as approaches infinity is , or in mathematical notation, $\lim_\frac = 2.$

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 In , a sequence is an enumerated collection of in which repetitions are allowed and matters. Like a , it contains (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unl ...

of
real number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s. One can state that the real number is the ''limit'' of this sequence, namely: :$\lim_ a_n = L$ which is read as :"The limit of ''an'' as ''n'' approaches infinity equals ''L''" if and only if :For every
real number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, there exists a
natural number File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...) In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, o ...
such that for all , we have . Intuitively, this means that eventually, all elements of the sequence get arbitrarily close to the limit, since the
absolute value of the absolute value function for real numbers In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calc ...

is the distance between and . Not every sequence has a limit; if it does, then it is called ''
convergent Convergent is an adjective for things that wikt:converge, converge. It is commonly used in mathematics and may refer to: *Convergent boundary, a type of plate tectonic boundary * Convergent (continued fraction) * Convergent evolution * Convergent s ...
'', and if it does not, then 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 as approaches infinity of a sequence is simply the limit at infinity of a function —defined on the
natural number File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...) In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, o ...
s . On the other hand, if is the domain of a function and if the limit as approaches infinity of is for ''every'' arbitrary sequence of points in which converges to , then the limit of the function as approaches is . One such sequence would be .

Limit as "standard part"

In
non-standard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard ...
(which involves a hyperreal number, hyperreal enlargement of the number system), the limit of a sequence $\left(a_n\right)$ can be expressed as the standard part function, standard part of the value $a_H$ of the natural extension of the sequence at an infinite hypernatural index ''n=H''. Thus, :$\lim_ a_n = \operatorname\left(a_H\right) .$ Here, the standard part function "st" rounds off each finite hyperreal number to the nearest real number (the difference between them is infinitesimal). This formalizes the natural intuition that for "very large" values of the index, the terms in the sequence are "very close" to the limit value of the sequence. Conversely, the standard part of a hyperreal $a=\left[a_n\right]$ represented in the ultrapower construction by a Cauchy sequence $\left(a_n\right)$, is simply the limit of that sequence: :$\operatorname\left(a\right)=\lim_ a_n .$ In this sense, taking the limit and taking the standard part are equivalent procedures.

Convergence and fixed point

A formal definition of convergence can be stated as follows. Suppose $p_n$ as $n$ goes from $0$ to $\infty$ is a sequence that converges to $p$, with $p_n \neq p$ for all $n$. If positive constants $\lambda$ and $\alpha$ exist with :$\lim_ \frac = \lambda$ then $p_n$ as $n$ goes from $0$ to $\infty$ converges to $p$ of order $\alpha$, with asymptotic error constant $\lambda$. Given a function $f$ with a fixed point $p$, there is a nice checklist for checking the convergence of the sequence $p_n$. # First check that p is indeed a fixed point: #:$f\left(p\right) = p$ # Check for linear convergence. Start by finding $\left , f\text{'} \left(p\right) \right ,$. If… #
• If it is found that there is something better than linear, the expression should be checked for quadratic convergence. Start by finding $\left, f\text{'}\text{'} \left(p\right) \$ If…
• Computability of the limit

Limits can be difficult to compute. There exist limit expressions whose modulus of convergence is Undecidable problem, undecidable. In recursion theory, the limit lemma proves that it is possible to encode undecidable problems using limits.''Recursively enumerable sets and degrees'', Soare, Robert I.

*Asymptotic analysis: a method of describing limiting behavior **Big O notation: used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity * Banach limit defined on the Banach space $\ell^\infty$ that extends the usual limits. * Cauchy sequence ** Complete metric space * Convergence of random variables * Convergent matrix * Limit (category theory), Limit in category theory **Direct limit **Inverse limit * Limit of a function ** One-sided limit: 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 theorem: finds a limit of a function via comparison with two other functions * Limit point * Limit set * Limit superior and limit inferior * Modes of convergence ** An Modes of convergence (annotated index), annotated index * Rate of convergence: the rate at which a convergent sequence approaches its limit