mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, the limit of a function is a fundamental concept in calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

and analysis
Analysis is the process of breaking a complex topic or substance
Substance may refer to:
* Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes
* Chemical substance, a material with a definite chemical composit ...

concerning the behavior of that 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 ...

near a particular input
Input may refer to:
Computing
* Input (computer science), the act of entering data into a computer or data processing system
* Information, any data entered into a computer or data processing system
* Input device
* Input method
* Input port (disam ...

.
Formal definitions, first devised in the early 19th century, are given below. Informally, a function ''f'' assigns an output
Output may refer to:
* The information produced by a computer, see Input/output
In computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of alg ...

''f''(''x'') to every input ''x''. We say that the function has a limit ''L'' at an input ''p,'' if ''f''(''x'') gets closer and closer to ''L'' as ''x'' moves closer and closer to ''p''. More specifically, when ''f'' is applied to any input ''sufficiently'' close to ''p'', the output value is forced ''arbitrarily'' close to ''L''. On the other hand, if some inputs very close to ''p'' are taken to outputs that stay a fixed distance apart, then we say the limit ''does not exist''.
The notion of a limit has many applications in . In particular, the many definitions of continuity employ the concept of limit: roughly, a function is continuous if all of its limits agree with the values of the function. The concept of limit also appears in the definition of the derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

: in the calculus of one variable, this is the limiting value of the slope
In mathematics, the slope or gradient of a line
Line, lines, The Line, or LINE may refer to:
Arts, entertainment, and media Films
* ''Lines'' (film), a 2016 Greek film
* ''The Line'' (2017 film)
* ''The Line'' (2009 film)
* ''The Line'', ...

of secant line
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

s to the graph of a function.
History

Although implicit in the development of calculus of the 17th and 18th centuries, the modern idea of the limit of a function goes back toBolzano
Bolzano ( or ; german: Bozen (formerly ), ; bar, Bozn; lld, Balsan or ) is the capital city
A capital or capital city is the municipality holding primary status in a Department (country subdivision), department, country, Constituent state, ...

who, in 1817, introduced the basics of the epsilon-delta technique to define continuous functions. However, his work was not known during his lifetime.
In his 1821 book ''Cours d'analyse
''Cours d'Analyse de l’École Royale Polytechnique; I.re Partie. Analyse algébrique'' is a seminal textbook in infinitesimal calculus published by Augustin-Louis Cauchy in 1821. The article follows the translation by Bradley and Sandifer in de ...

'', Cauchy
Baron Augustin-Louis Cauchy (; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was ...

discussed variable quantities, infinitesimal
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s and limits, and defined continuity of $y=f(x)$ by saying that an infinitesimal change in ''x'' necessarily produces an infinitesimal change in ''y'', while claims that he used a rigorous epsilon-delta definition in proofs., collected iWho Gave You the Epsilon?

pp. 5–13. Also available at: http://www.maa.org/pubs/Calc_articles/ma002.pdf In 1861,

Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) incl ...

first introduced the epsilon-delta definition of limit in the form it is usually written today. He also introduced the notations lim and limHardy
Hardy may refer to:
People
* Hardy (surname)
* Hardy (given name)
* Hardy (singer), American singer-songwriter Places Antarctica
* Mount Hardy, Enderby Land
* Hardy Cove, Greenwich Island
* Hardy Rocks, Biscoe Islands
Australia
* Hardy, South A ...

, which is introduced in his book ''A Course of Pure Mathematics
''A Course of Pure Mathematics'' is a classic textbook in introductory mathematical analysis, written by G. H. Hardy. It is recommended for people studying calculus. First published in 1908, it went through ten editions (up to 1952) and several r ...

'' in 1908.
Motivation

Imagine a person walking over a landscape represented by the graph of ''y'' = ''f''(''x''). Their horizontal position is measured by the value of ''x'', much like the position given by a map of the land or by aglobal positioning system
The Global Positioning System (GPS), originally Navstar GPS, is a satellite-based radionavigation system owned by the United States government
The federal government of the United States (U.S. federal government or U.S. governme ...

. Their altitude is given by the coordinate ''y''. They walk toward the horizontal position given by ''x'' = ''p''. As they get closer and closer to it, they notice that their altitude approaches ''L''. If asked about the altitude of ''x'' = ''p'', they would then answer ''L''.
What, then, does it mean to say, their altitude is approaching ''L?'' It means that their altitude gets nearer and nearer to ''L''—except for a possible small error in accuracy. For example, suppose we set a particular accuracy goal for our traveler: they must get within ten meters of ''L''. They report back that indeed, they can get within ten vertical meters of ''L'', since they note that when they are within fifty horizontal meters of ''p'', their altitude is ''always'' ten meters or less from ''L''.
The accuracy goal is then changed: can they get within one vertical meter? Yes. If they are anywhere within seven horizontal meters of ''p'', their altitude will always remain within one meter from the target ''L''. In summary, to say that the traveler's altitude approaches ''L'' as their horizontal position approaches ''p'', is to say that for every target accuracy goal, however small it may be, there is some neighbourhood of ''p'' whose altitude fulfills that accuracy goal.
The initial informal statement can now be explicated:
:The limit of a function ''f''(''x'') as ''x'' approaches ''p'' is a number ''L'' with the following property: given any target distance from ''L'', there is a distance from ''p'' within which the values of ''f''(''x'') remain within the target distance.
In fact, this explicit statement is quite close to the formal definition of the limit of a function, with values in a topological space
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 gener ...

.
More specifically, to say that
:$\backslash lim\_f(x)\; =\; L,\; \backslash ,$
is to say that ''ƒ''(''x'') can be made as close to ''L'' as desired, by making ''x'' close enough, but not equal, to ''p''.
The following definitions, known as (ε, δ)-definitions, are the generally accepted definitions for the limit of a function in various contexts.
Functions of a single variable

(ε, δ)-definition of limit

Suppose ''f'' : R → R is defined on thereal line
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

and . One would say that the limit of ''f'', as ''x'' approaches ''p'', is ''L'' and written
:$\backslash lim\_f(x)\; =\; L,$
or alternatively as:
:$f(x)\; \backslash to\; L$ as $x\; \backslash to\; p$ (reads "$f(x)$ tends to $L$ as $x$ tends to $p$")
if the following property holds:
* For every real , there exists a real such that for all real x, implies that .
A more general definition applies for functions defined on subset
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s of the real line. Let (''a'', ''b'') be an open interval
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 ge ...

in R, and ''p'' a point of (''a'', ''b''). Let ''f'' be a real-valued function
In mathematics, a real-valued function is 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 ari ...

defined on all of (''a'', ''b'')—except possibly at ''p'' itself. It is then said that the limit of ''f'' as ''x'' approaches ''p'' is ''L,'' if for every real , there exists a real such that and implies that .
Here, note that the value of the limit does not depend on ''f'' being defined at ''p'', nor on the value ''f''(''p'')—if it is defined.
The letters ''ε'' and ''δ'' can be understood as "error" and "distance". In fact, Cauchy used ''ε'' as an abbreviation for "error" in some of his work, though in his definition of continuity, he used an infinitesimal $\backslash alpha$ rather than either ''ε'' or ''δ'' (see ''Cours d'Analyse
''Cours d'Analyse de l’École Royale Polytechnique; I.re Partie. Analyse algébrique'' is a seminal textbook in infinitesimal calculus published by Augustin-Louis Cauchy in 1821. The article follows the translation by Bradley and Sandifer in de ...

''). In these terms, the error (''ε'') in the measurement of the value at the limit can be made as small as desired, by reducing the distance (''δ'') to the limit point. As discussed below, this definition also works for functions in a more general context. The idea that ''δ'' and ''ε'' represent distances helps suggest these generalizations.
Existence and one-sided limits

Alternatively, ''x'' may approach ''p'' from above (right) or below (left), in which case the limits may be written as :$\backslash lim\_f(x)\; =\; L$ or :$\backslash lim\_f(x)\; =\; L$ respectively. If these limits exist at p and are equal there, then this can be referred to as ''the'' limit of ''f''(''x'') at ''p''. If the one-sided limits exist at ''p'', but are unequal, then there is no limit at ''p'' (i.e., the limit at ''p'' does not exist). If either one-sided limit does not exist at ''p'', then the limit at ''p'' also does not exist. A formal definition is as follows. The limit of ''f''(''x'') as ''x'' approaches ''p'' from above is ''L'' if, for every ''ε'' > 0, there exists a ''δ'' > 0 such that , ''f''(''x'') − ''L'', < ''ε'' whenever 0 < ''x'' − ''p'' < ''δ''. The limit of ''f''(''x'') as ''x'' approaches ''p'' from below is ''L'' if, for every ''ε'' > 0, there exists a ''δ'' > 0 such that , ''f''(''x'') − ''L'', < ''ε'' whenever 0 < ''p'' − ''x'' < ''δ''. If the limit does not exist, then theoscillation
Oscillation is the repetitive variation, typically in time, of some measure about a central value (often a point of Mechanical equilibrium, equilibrium) or between two or more different states. The term ''vibration'' is precisely used to describ ...

of ''f'' at ''p'' is non-zero.
More general subsets

Apart from open intervals, limits can be defined for functions on arbitrary subsets of R, as follows : let ''f'' be a real-valued function defined on a subset ''S'' of the real line. Let ''p'' be alimit point
In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is def ...

of ''S''—that is, ''p'' is the limit of some sequence of elements of ''S'' distinct from p. The limit of ''f'', as ''x'' approaches ''p'' from values in ''S'', is ''L,'' if for every , there exists a such that and implies that .
This limit is often written as:
:$L\; =\; \backslash lim\_\; f(x).$
The condition that ''f'' be defined on ''S'' is that ''S'' be a subset of the domain of ''f''. This generalization includes as special cases limits on an interval, as well as left-handed limits of real-valued functions (e.g., by taking ''S'' to be an open interval of the form $(-\backslash infty,a)$), and right-handed limits (e.g., by taking ''S'' to be an open interval of the form $(a,\backslash infty)$). It also extends the notion of one-sided limits to the included endpoints of (half-)closed intervals, so the square root function
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) ...

''f''(''x'')= can have limit 0 as x approaches 0 from above.
Deleted versus non-deleted limits

The definition of limit given here does not depend on how (or whether) ''f'' is defined at ''p''. refers to this as a ''deleted limit'', because it excludes the value of ''f'' at ''p''. The corresponding ''non-deleted limit'' does depend on the value of ''f'' at ''p'', if ''p'' is in the domain of ''f'': * A number ''L'' is the non-deleted limit of ''f'' as ''x'' approaches ''p'' if, for every ''ε'' > ''0'', there exists a ''δ'' > ''0'' such that , ''x'' − ''p'' , < ''δ'' and implies , ''f''(''x'') − ''L'' , < ''ε''. The definition is the same, except that the neighborhood , ''x'' − ''p'' , < ''δ'' now includes the point ''p'', in contrast to the deleted neighborhood 0 < , ''x'' − ''p'' , < ''δ''. This makes the definition of a non-deleted limit less general. One of the advantages of working with non-deleted limits is that they allow to state the theorem about limits of compositions without any constraints on the functions (other than the existence of their non-deleted limits) (). notes that although by "limit" some authors do mean this non-deleted limit, deleted limits are the most popular. For example, , , , , all take "limit" to mean the deleted limit.Examples

Non-existence of one-sided limit(s)

The function :$f(x)=\backslash begin\; \backslash sin\backslash frac\; \&\; \backslash text\; x<1\; \backslash \backslash \; 0\; \&\; \backslash text\; x=1\; \backslash \backslash \; \backslash frac\&\; \backslash text\; x>1\; \backslash end$ has no limit at $x\_0\; =\; 1$ (the left-hand limit does not exist due to the oscillatory nature of the sine function, and the right-hand limit does not exist due to the asymptotic behaviour of the reciprocal function), but has a limit at every other ''x''-coordinate. The function :$f(x)=\backslash begin\; 1\; \&\; x\; \backslash text\; \backslash \backslash \; 0\; \&\; x\; \backslash text\; \backslash end$ (a.k.a., theDirichlet function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

) has no limit at any ''x''-coordinate.
Non-equality of one-sided limits

The function :$f(x)=\backslash begin\; 1\; \&\; \backslash text\; x\; <\; 0\; \backslash \backslash \; 2\; \&\; \backslash text\; x\; \backslash ge\; 0\; \backslash end$ has a limit at every non-zero ''x''-coordinate (the limit equals 1 for negative ''x'' and equals 2 for positive ''x''). The limit at ''x'' = 0 does not exist (the left-hand limit equals 1, whereas the right-hand limit equals 2).Limits at only one point

The functions :$f(x)=\backslash begin\; x\; \&\; x\; \backslash text\; \backslash \backslash \; 0\; \&\; x\; \backslash text\; \backslash end$ and :$f(x)=\backslash begin\; ,\; x,\; \&\; x\; \backslash text\; \backslash \backslash \; 0\; \&\; x\; \backslash text\; \backslash end$ both have a limit at ''x'' = 0 and it equals 0.Limits at countably many points

The function :$f(x)=\backslash begin\; \backslash sin\; x\; \&\; x\; \backslash text\; \backslash \backslash \; 1\; \&\; x\; \backslash text\; \backslash end$ has a limit at any ''x''-coordinate of the form $\backslash frac\; +\; 2n\backslash pi$, where ''n'' is any integer.Functions on metric spaces

Suppose ''M'' and ''N'' are subsets ofmetric spaces
Metric or metrical may refer to:
* Metric system
The metric system is a that succeeded the decimalised system based on the introduced in France in the 1790s. The historical development of these systems culminated in the definition of the ...

''A'' and ''B'', respectively, and ''f'' : ''M'' → ''N'' is defined between ''M'' and ''N'', with ''x'' ∈ ''M,'' ''p'' a limit point
In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is def ...

of ''M'' and ''L'' ∈ ''N''. It is said that the limit of ''f'' as ''x'' approaches ''p'' is ''L'' and write
:$\backslash lim\_f(x)\; =\; L$
if the following property holds:
* For every ε > 0, there exists a δ > 0 such that dneighbourhood
A neighbourhood (British English, Hiberno-English, Hibernian English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographicall ...

is as follows:
:$\backslash lim\_f(x)\; =\; L$
if, for every neighbourhood ''V'' of ''L'' in ''B'', there exists a neighbourhood ''U'' of ''p'' in ''A'' such that ''f''(U ∩ M − ) ⊆ ''V''.
Functions on topological spaces

Suppose ''X'',''Y'' aretopological space
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 gener ...

s with ''Y'' a Hausdorff space
In topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...

. Let ''p'' be a limit point
In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is def ...

of Ω ⊆ ''X'', and ''L'' ∈''Y''. For a function ''f'' : Ω → ''Y'', it is said that the limit of ''f'' as ''x'' approaches ''p'' is ''L'' (i.e., ''f''(''x'') → ''L'' as ''x'' → ''p'') and written
:$\backslash lim\_f(x)\; =\; L$
if the following property holds:
* For every open neighborhood
A neighbourhood (British English
British English (BrE) is the standard dialect
A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...

''V'' of ''L'', there exists an open neighborhood ''U'' of ''p'' such that ''f''(''U'' ∩ Ω − ) ⊆ ''V''.
This last part of the definition can also be phrased "there exists an open punctured neighbourhood ''U'' of ''p'' such that ''f''(''U''∩Ω) ⊆ ''V'' ".
Note that the domain of ''f'' does not need to contain ''p''. If it does, then the value of ''f'' at ''p'' is irrelevant to the definition of the limit. In particular, if the domain of ''f'' is ''X'' − (or all of ''X''), then the limit of ''f'' as ''x'' → ''p'' exists and is equal to ''L'' if, for all subsets Ω of ''X'' with limit point ''p'', the limit of the restriction of ''f'' to Ω exists and is equal to ''L''. Sometimes this criterion is used to establish the ''non-existence'' of the two-sided limit of a function on R by showing that the one-sided limit
In calculus, a one-sided limit is either of the two Limit of a function, limits of a function (mathematics), function ''f''(''x'') of a real number, real variable ''x'' as ''x'' approaches a specified point either from the left or from the right.
...

s either fail to exist or do not agree. Such a view is fundamental in the field of general topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, where limits and continuity at a point are defined in terms of special families of subsets, called filters
Filter, filtering or filters may refer to:
Science and technology Device
* Filter (chemistry)
Filtration is a physical separation process
A separation process is a method that converts a mixture or solution of chemical substances into two o ...

, or generalized sequences known as nets.
Alternatively, the requirement that ''Y'' be a Hausdorff space can be relaxed to the assumption that ''Y'' be a general topological space, but then the limit of a function may not be unique. In particular, one can no longer talk about ''the limit'' of a function at a point, but rather ''a limit'' or ''the set of limits'' at a point.
A function is continuous at a limit point ''p'' of and in its domain if and only if ''f''(''p'') is ''the'' (or, in the general case, ''a'') limit of ''f''(''x'') as ''x'' tends to ''p''.
Limits involving infinity

Limits at infinity

Let $S\backslash subseteq\backslash mathbb$, $x\backslash in\; S$ and $f:S\backslash mapsto\backslash mathbb$. The limit of ''f'' as ''x'' approaches infinity is ''L'', denoted :$\backslash lim\_f(x)\; =\; L,$ means that for all $\backslash varepsilon\; >\; 0$, there exists ''c'' such that $,\; f(x)\; -\; L,\; <\; \backslash varepsilon$ whenever ''x'' > ''c''. Or, symbolically: :$\backslash forall\; \backslash varepsilon\; >\; 0\; \backslash ;\; \backslash exists\; c\; \backslash ;\; \backslash forall\; x\; >\; c\; :\backslash ;\; ,\; f(x)\; -\; L,\; <\; \backslash varepsilon$. Similarly, the limit of ''f'' as ''x'' approaches negative infinity is ''L'', denoted :$\backslash lim\_f(x)\; =\; L,$ means that for all $\backslash varepsilon\; >\; 0$ there exists ''c'' such that $,\; f(x)\; -\; L,\; <\; \backslash varepsilon$ whenever ''x'' < ''c''. Or, symbolically: :$\backslash forall\; \backslash varepsilon\; >\; 0\; \backslash ;\; \backslash exists\; c\; \backslash ;\; \backslash forall\; x\; <\; c\; :\backslash ;\; ,\; f(x)\; -\; L,\; <\; \backslash varepsilon$. For example, :$\backslash lim\_e^x\; =\; 0.\; \backslash ,$Infinite limits

For a function whose values grow without bound, the function diverges and the usual limit does not exist. However, in this case one may introduce limits with infinite values. Let $S\backslash subseteq\backslash mathbb$, $x\backslash in\; S$ and $f:S\backslash mapsto\backslash mathbb$. The statement the limit of ''f'' as ''x'' approaches ''a'' is infinity, denoted :$\backslash lim\_\; f(x)\; =\; \backslash infty,$ means that for all $N\; >\; 0$ there exists $\backslash delta\; >\; 0$ such that $f(x)\; >\; N$ whenever $0\; <\; ,\; x\; -\; a,\; <\; \backslash delta.$ These ideas can be combined in a natural way to produce definitions for different combinations, such as :$\backslash lim\_\; f(x)\; =\; \backslash infty,\; \backslash lim\_f(x)\; =\; -\backslash infty.$ For example, :$\backslash lim\_\; \backslash ln\; x\; =\; -\backslash infty.$ Limits involving infinity are connected with the concept ofasymptote
In analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέ ...

s.
These notions of a limit attempt to provide a metric space interpretation to limits at infinity. In fact, they are consistent with the topological space definition of limit if
*a neighborhood of −∞ is defined to contain an interval ∞, ''c'') for some ''c'' ∈ R,
*a neighborhood of ∞ is defined to contain an interval (''c'', ∞where ''c'' ∈ R, and
*a neighborhood of ''a'' ∈ R is defined in the normal way metric space R.
In this case, R is a topological space and any function of the form ''f'': ''X'' → ''Y'' with ''X'', ''Y''⊆ R is subject to the topological definition of a limit. Note that with this topological definition, it is easy to define infinite limits at finite points, which have not been defined above in the metric sense.
Alternative notation

Many authors allow for theprojectively extended real line
Image:Real projective line.svg,
The projectively extended real line can be visualized as the real number line wrapped around a circle (by some form of stereographic projection) with an additional point at infinity.
In real analysis, the projectiv ...

to be used as a way to include infinite values as well as extended real line. With this notation, the extended real line is given as and the projectively extended real line is R ∪ where a neighborhood of ∞ is a set of the form The advantage is that one only needs three definitions for limits (left, right, and central) to cover all the cases.
As presented above, for a completely rigorous account, we would need to consider 15 separate cases for each combination of infinities (five directions: −∞, left, central, right, and +∞; three bounds: −∞, finite, or +∞). There are also noteworthy pitfalls. For example, when working with the extended real line, $x^$ does not possess a central limit (which is normal):
:$\backslash lim\_\; =\; +\backslash infty,\; \backslash lim\_\; =\; -\backslash infty.$
In contrast, when working with the projective real line, infinities (much like 0) are unsigned, so, the central limit ''does'' exist in that context:
:$\backslash lim\_\; =\; \backslash lim\_\; =\; \backslash lim\_\; =\; \backslash infty.$
In fact there are a plethora of conflicting formal systems in use.
In certain applications of numerical differentiation and integration, it is, for example, convenient to have signed zeroes.
A simple reason has to do with the converse of $\backslash lim\_\; =\; -\backslash infty$, namely, it is convenient for $\backslash lim\_\; =\; -0$ to be considered true.
Such zeroes can be seen as an approximation to infinitesimal
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s.
Limits at infinity for rational functions

There are three basic rules for evaluating limits at infinity for arational function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

''f''(''x'') = ''p''(''x'')/''q''(''x''): (where ''p'' and ''q'' are polynomials):
*If the degree
Degree may refer to:
As a unit of measurement
* Degree symbol (°), a notation used in science, engineering, and mathematics
* Degree (angle), a unit of angle measurement
* Degree (temperature), any of various units of temperature measurement ...

of ''p'' is greater than the degree of ''q'', then the limit is positive or negative infinity depending on the signs of the leading coefficients;
*If the degree of ''p'' and ''q'' are equal, the limit is the leading coefficient of ''p'' divided by the leading coefficient of ''q'';
*If the degree of ''p'' is less than the degree of ''q'', the limit is 0.
If the limit at infinity exists, it represents a horizontal asymptote at ''y'' = ''L''. Polynomials do not have horizontal asymptotes; such asymptotes may however occur with rational functions.
Functions of more than one variable

By noting that , ''x'' − ''p'', represents a distance, the definition of a limit can be extended to functions of more than one variable. In the case of a function ''f'' : REuclidean distance
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

. This can be extended to any number of variables.
Sequential limits

Let be a mapping from a topological space ''X'' into a Hausdorff space ''Y'', a limit point of ''X'' and . :The sequential limit of ''f'' as ''x'' tends to ''p'' is ''L'' if, for everysequence
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

(''x''metrizable
In topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...

, then ''L'' is the sequential limit of ''f'' as ''x'' approaches ''p'' if and only if it is the limit (in the sense above) of ''f'' as ''x'' approaches ''p''.
Other characterizations

In terms of sequences

For functions on the real line, one way to define the limit of a function is in terms of the limit of sequences. (This definition is usually attributed toEduard Heine
Heinrich Eduard Heine (16 March 1821 – October 1881) was a German
German(s) may refer to:
Common uses
* of or related to Germany
* Germans, Germanic ethnic group, citizens of Germany or people of German ancestry
* For citizens of German ...

.) In this setting:
:$\backslash lim\_f(x)=L$
if, and only if, for all sequences $x\_n$ (with $x\_n$ not equal to ''a'' for all ''n'') converging to $a$ the sequence $f(x\_n)$ converges to $L$. It was shown by Sierpiński in 1916 that proving the equivalence of this definition and the definition above, requires and is equivalent to a weak form of the axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

. Note that defining what it means for a sequence $x\_n$ to converge to $a$ requires the epsilon, delta method.
Similarly as it was the case of Weierstrass's definition, a more general Heine definition applies to functions defined on subset
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s of the real line. Let ''f'' be a real-valued function with the domain ''Dm''(''f''). Let ''a'' be the limit of a sequence of elements of ''Dm''(''f'') \ . Then the limit (in this sense) of ''f'' is ''L'' as ''x'' approaches ''p''
if for every sequence $x\_n$ ∈ ''Dm''(''f'') \ (so that for all ''n'', $x\_n$ is not equal to ''a'') that converges to ''a'', the sequence $f(x\_n)$ converges to $L$. This is the same as the definition of a sequential limit in the preceding section obtained by regarding the subset ''Dm''(''f'') of R as a metric space with the induced metric.
In non-standard calculus

In non-standard calculus the limit of a function is defined by: :$\backslash lim\_f(x)=L$ if and only if for all $x\backslash in\; \backslash mathbb^*$, $f^*(x)-L$ is infinitesimal whenever $x-a$ is infinitesimal. Here $\backslash mathbb^*$ are thehyperreal 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 and $f^*$ is the natural extension of ''f'' to the non-standard real numbers. Keisler proved that such a hyperreal reduces the quantifier complexity by two quantifiers. On the other hand, Hrbacek writes that for the definitions to be valid for all hyperreal numbers they must implicitly be grounded in the ε-δ method, and claims that, from the pedagogical point of view, the hope that non-standard calculus could be done without ε-δ methods cannot be realized in full.
Bŀaszczyk et al. detail the usefulness of microcontinuity In nonstandard analysis, a discipline within classical mathematics, microcontinuity (or ''S''-continuity) of an internal function ''f'' at a point ''a'' is defined as follows:
:for all ''x'' infinitely close to ''a'', the value ''f''(''x'') is infin ...

in developing a transparent definition of uniform continuity, and characterize Hrbacek's criticism as a "dubious lament".
In terms of nearness

At the 1908 international congress of mathematics F. Riesz introduced an alternate way defining limits and continuity in concept called "nearness". A point $x$ is defined to be near a set $A\backslash subseteq\; \backslash mathbb$ if for every $r>0$ there is a point $a\backslash in\; A$ so that $,\; x-a,math>.\; In\; this\; setting\; the\; :$ \backslash lim\_\; f(x)=L$if\; and\; only\; if\; for\; all$ A\backslash subseteq\; \backslash mathbb$,$ L$is\; near$ f(A)$whenever$ a$is\; near$ A$.\; Here$ f(A)$is\; the\; set$ \backslash $.\; This\; definition\; can\; also\; be\; extended\; to\; metric\; and\; topological\; spaces.$Relationship to continuity

The notion of the limit of a function is very closely related to the concept of continuity. A function ''ƒ'' is said to becontinuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ga ...

at ''c'' if it is both defined at ''c'' and its value at ''c'' equals the limit of ''f'' as ''x'' approaches ''c'':
: $\backslash lim\_\; f(x)\; =\; f(c).$
(We have here assumed that ''c'' is a Properties

If a function ''f'' is real-valued, then the limit of ''f'' at ''p'' is ''L'' if and only if both the right-handed limit and left-handed limit of ''f'' at ''p'' exist and are equal to ''L''. The function ''f'' iscontinuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ga ...

at ''p'' if and only if the limit of ''f''(''x'') as ''x'' approaches ''p'' exists and is equal to ''f''(''p''). If ''f'' : ''M'' → ''N'' is a function between metric spaces ''M'' and ''N'', then it is equivalent that ''f'' transforms every sequence in ''M'' which converges towards ''p'' into a sequence in ''N'' which converges towards ''f''(''p'').
If ''N'' is a normed vector space
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 g ...

, then the limit operation is linear in the following sense: if the limit of ''f''(''x'') as ''x'' approaches ''p'' is ''L'' and the limit of ''g''(''x'') as ''x'' approaches ''p'' is ''P'', then the limit of ''f''(''x'') + g(''x'') as ''x'' approaches ''p'' is ''L'' + ''P''. If ''a'' is a scalar from the base field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

, then the limit of ''af''(''x'') as ''x'' approaches ''p'' is ''aL''.
If ''f'' and ''g'' are real-valued (or complex-valued) functions, then taking the limit of an operation on ''f''(''x'') and ''g''(''x'') (e.g., $f+g$'','' $f-g$'','' $f\backslash times\; g$'','' $f/g$'','' $f^g$) under certain conditions is compatible with the operation of limits of ''f(x)'' and ''g(x)''. This fact is often called the algebraic limit theorem. The main condition needed to apply the following rules is that the limits on the right-hand sides of the equations exist (in other words, these limits are finite values including 0). Additionally, the identity for division requires that the denominator on the right-hand side is non-zero (division by 0 is not defined), and the identity for exponentiation requires that the base is positive, or zero while the exponent is positive (finite).
:$\backslash begin\; \backslash lim\backslash limits\_\; \&\; (f(x)\; +\; g(x))\; \&\; =\; \&\; \backslash lim\backslash limits\_\; f(x)\; +\; \backslash lim\backslash limits\_\; g(x)\; \backslash \backslash \; \backslash lim\backslash limits\_\; \&\; (f(x)\; -\; g(x))\; \&\; =\; \&\; \backslash lim\backslash limits\_\; f(x)\; -\; \backslash lim\backslash limits\_\; g(x)\; \backslash \backslash \; \backslash lim\backslash limits\_\; \&\; (f(x)\backslash cdot\; g(x))\; \&\; =\; \&\; \backslash lim\backslash limits\_\; f(x)\; \backslash cdot\; \backslash lim\backslash limits\_\; g(x)\; \backslash \backslash \; \backslash lim\backslash limits\_\; \&\; (f(x)/g(x))\; \&\; =\; \&\; \backslash \backslash \; \backslash lim\backslash limits\_\; \&\; f(x)^\; \&\; =\; \&\; \backslash end$
These rules are also valid for one-sided limits, including when ''p'' is ∞ or −∞. In each rule above, when one of the limits on the right is ∞ or −∞, the limit on the left may sometimes still be determined by the following rules.
*''q'' + ∞ = ∞ if ''q'' ≠ −∞
*''q'' × ∞ = ∞ if ''q'' > 0
*''q'' × ∞ = −∞ if ''q'' < 0
*''q'' / ∞ = 0 if ''q'' ≠ ∞ and ''q'' ≠ −∞
*∞Extended real number line
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

).
In other cases the limit on the left may still exist, although the right-hand side, called an ''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 ...

'', does not allow one to determine the result. This depends on the functions ''f'' and ''g''. These indeterminate forms are:
* 0 / 0
* ±∞ / ±∞
* 0 × ±∞
* ∞ + −∞
* 0L'Hôpital's rule
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

below and 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 ...

.
Limits of compositions of functions

In general, from knowing that :$\backslash lim\_\; f(y)\; =\; c$ and $\backslash lim\_\; g(x)\; =\; b$, it does ''not'' follow that $\backslash lim\_\; f(g(x))\; =\; c$. However, this "chain rule" does hold if one of the following ''additional'' conditions holds: *''f''(''b'') = ''c'' (that is, ''f'' is continuous at ''b''), or *''g'' does not take the value ''b'' near ''a'' (that is, there exists a $\backslash delta\; >0$ such that if $0<,\; x-a,\; <\backslash delta$ then $,\; g(x)-b,\; >0$). As an example of this phenomenon, consider the following functions that violates both additional restrictions: :$f(x)=g(x)=\backslash begin0\; \&\; \backslash text\; x\backslash neq\; 0\; \backslash \backslash \; 1\; \&\; \backslash text\; x=0\; \backslash end.$ Since the value at ''f''(0) is a removable discontinuity, :$\backslash lim\_\; f(x)\; =\; 0$ for all $a$. Thus, the naïve chain rule would suggest that the limit of ''f''(''f''(''x'')) is 0. However, it is the case that :$f(f(x))=\backslash begin1\; \&\; \backslash text\; x\backslash neq\; 0\; \backslash \backslash \; 0\; \&\; \backslash text\; x=0\; \backslash end$ and so :$\backslash lim\_\; f(f(x))\; =\; 1$ for all $a$.Limits of special interest

Rational functions

For $n$ a nonnegative integer and constants $a\_1,\; a\_2,\; a\_3,\backslash ldots,\; a\_n$ and $b\_1,\; b\_2,\; b\_3,\backslash ldots,\; b\_n$, *$\backslash lim\_\; \backslash frac\; =\; \backslash frac$ This can be proven by dividing both the numerator and denominator by $x^$. If the numerator is a polynomial of higher degree, the limit does not exist. If the denominator is of higher degree, the limit is 0.Trigonometric functions

*$\backslash lim\_\; \backslash frac\; =\; 1$ *$\backslash lim\_\; \backslash frac\; =\; 0$Exponential functions

*$\backslash lim\_\; (1+x)^\; =\; \backslash lim\_\; \backslash left(1+\backslash frac\backslash right)^\; =\; e$ *$\backslash lim\_\; \backslash frac\; =\; 1$ *$\backslash lim\_\; \backslash frac\; =\; \backslash frac$ *$\backslash lim\_\; \backslash frac\; =\; \backslash frac\backslash ln\; c$ *$\backslash lim\_\; x^\; =\; 1$Logarithmic functions

*$\backslash lim\_\; \backslash frac\; =\; 1$ *$\backslash lim\_\; \backslash frac\; =\; \backslash frac$ *$\backslash lim\_\; \backslash frac\; =\; \backslash frac$L'Hôpital's rule

This rule usesderivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

s to find limits of indeterminate forms
In 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. Th ...

or , and only applies to such cases. Other indeterminate forms may be manipulated into this form. Given two functions and , defined over an open interval
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 ge ...

containing the desired limit point ''c'', then if:
# $\backslash lim\_f(x)=\backslash lim\_g(x)=0,$ or $\backslash lim\_f(x)=\backslash pm\backslash lim\_g(x)\; =\; \backslash pm\backslash infty$, and
# $f$ and $g$ are differentiable over $I\; \backslash setminus\; \backslash $, and
# $g\text{'}(x)\backslash neq\; 0$ for all $x\; \backslash in\; I\; \backslash setminus\; \backslash $, and
# $\backslash lim\_\backslash frac$ exists,
then:
$\backslash lim\_\; \backslash frac\; =\; \backslash lim\_\; \backslash frac$
Normally, the first condition is the most important one.
For example:
$\backslash lim\_\; \backslash frac\; =\; \backslash lim\_\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac.$
Summations and integrals

Specifying an infinite bound on a summation or integral is a common shorthand for specifying a limit. A short way to write the limit $\backslash lim\_\; \backslash sum\_^n\; f(i)$ is $\backslash sum\_^\backslash infty\; f(i)$. An important example of limits of sums such as these areseries
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used i ...

.
A short way to write the limit $\backslash lim\_\; \backslash int\_a^x\; f(t)\; \backslash ;\; dt$
is $\backslash int\_a^\backslash infty\; f(t)\; \backslash ;\; dt$.
A short way to write the limit $\backslash lim\_\; \backslash int\_x^b\; f(t)\; \backslash ;\; dt$
is $\backslash int\_^b\; f(t)\; \backslash ;\; dt$.
See also

* * * * * * * * *Notes

References

* * * * * * . * * * *External links

MacTutor History of Weierstrass.

Visual Calculus

by Lawrence S. Husch,

University of Tennessee
The University of Tennessee (University of Tennessee, Knoxville; UT Knoxville; UTK; or UT) is a public
In public relations
Public relations (PR) is the practice of managing and disseminating information from an individual or an organ ...

(2001)
{{DEFAULTSORT:Limit Of A Function
Limits (mathematics)
Functions and mappings