TheInfoList

Although the function (sin ''x'')/''x'' is not defined at zero, as ''x'' becomes closer and closer to zero, (sin ''x'')/''x'' becomes arbitrarily close to 1. In other words, the limit of (sin ''x'')/''x'', as ''x'' approaches zero, equals 1.
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 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
modern calculus . 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 to
Bolzano 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 $y=f\left(x\right)$ 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 i
Who 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 lim''x''→''x''0. The modern notation of placing the arrow below the limit symbol is due to
Hardy 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 a
global 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 :$\lim_f\left(x\right) = L, \,$ 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 the
real 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 :$\lim_f\left(x\right) = L,$ or alternatively as: :$f\left(x\right) \to L$ as $x \to p$ (reads "$f\left(x\right)$ 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 $\alpha$ 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 :$\lim_f\left(x\right) = L$ or :$\lim_f\left(x\right) = L$ 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 the
oscillation 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 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 ''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: :$L = \lim_ f\left(x\right).$ 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 $\left(-\infty,a\right)$), and right-handed limits (e.g., by taking ''S'' to be an open interval of the form $\left(a,\infty\right)$). 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 :$f\left(x\right)=\begin \sin\frac & \text x<1 \\ 0 & \text x=1 \\ \frac& \text x>1 \end$ 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 :$f\left(x\right)=\begin 1 & x \text \\ 0 & x \text \end$ (a.k.a., the
Dirichlet 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 :$f\left(x\right)=\begin 1 & \text x < 0 \\ 2 & \text x \ge 0 \end$ 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 :$f\left(x\right)=\begin x & x \text \\ 0 & x \text \end$ and :$f\left(x\right)=\begin , x, & x \text \\ 0 & x \text \end$ both have a limit at ''x'' = 0 and it equals 0.

### Limits at countably many points

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

# Functions on metric spaces

Suppose ''M'' and ''N'' are subsets of
metric 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 :$\lim_f\left(x\right) = L$ if the following property holds: * For every ε > 0, there exists a δ > 0 such that d''B''(''f''(''x''), ''L'') < ε whenever 0 < ''d''''A''(''x'', ''p'') < ''δ''. Again, note that ''p'' need not be in the domain of ''f'', nor does ''L'' need to be in the range of ''f'', and even if ''f''(''p'') is defined it need not be equal to ''L''. An alternative definition using the concept of
neighbourhood 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: :$\lim_f\left(x\right) = L$ 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'' are
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 ...
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 :$\lim_f\left(x\right) = L$ 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 $S\subseteq\mathbb$, $x\in S$ and $f:S\mapsto\mathbb$. The limit of ''f'' as ''x'' approaches infinity is ''L'', denoted :$\lim_f\left(x\right) = L,$ means that for all $\varepsilon > 0$, there exists ''c'' such that $, f\left(x\right) - L, < \varepsilon$ whenever ''x'' > ''c''. Or, symbolically: :$\forall \varepsilon > 0 \; \exists c \; \forall x > c :\; , f\left(x\right) - L, < \varepsilon$. Similarly, the limit of ''f'' as ''x'' approaches negative infinity is ''L'', denoted :$\lim_f\left(x\right) = L,$ means that for all $\varepsilon > 0$ there exists ''c'' such that $, f\left(x\right) - L, < \varepsilon$ whenever ''x'' < ''c''. Or, symbolically: :$\forall \varepsilon > 0 \; \exists c \; \forall x < c :\; , f\left(x\right) - L, < \varepsilon$. For example, :$\lim_e^x = 0. \,$

## 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 $S\subseteq\mathbb$, $x\in S$ and $f:S\mapsto\mathbb$. The statement the limit of ''f'' as ''x'' approaches ''a'' is infinity, denoted :$\lim_ f\left(x\right) = \infty,$ means that for all $N > 0$ there exists $\delta > 0$ such that $f\left(x\right) > N$ whenever $0 < , x - a, < \delta.$ These ideas can be combined in a natural way to produce definitions for different combinations, such as :$\lim_ f\left(x\right) = \infty, \lim_f\left(x\right) = -\infty.$ For example, :$\lim_ \ln x = -\infty.$ Limits involving infinity are connected with the concept of
asymptote 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 the
projectively 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): :$\lim_ = +\infty, \lim_ = -\infty.$ In contrast, when working with the projective real line, infinities (much like 0) are unsigned, so, the central limit ''does'' exist in that context: :$\lim_ = \lim_ = \lim_ = \infty.$ 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 $\lim_ = -\infty$, namely, it is convenient for $\lim_ = -0$ 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 a
rational 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'' : R2 → R, :$\lim_ f\left(x, y\right) = L$ if :for every ''ε'' > 0 there exists a δ > 0 such that for all (''x'',''y'') with 0 < , , (''x'',''y'') − (''p'',''q''), , < δ, then , ''f''(''x'',''y'') − ''L'', < ε where , , (''x'',''y'') − (''p'',''q''), , represents the
Euclidean 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 every
sequence 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''''n'') in that converges to ''p'', the sequence ''f''(''x''''n'') converges to ''L''. If ''L'' is the limit (in the sense above) of ''f'' as ''x'' approaches ''p'', then it is a sequential limit as well, however the converse need not hold in general. If in addition ''X'' is
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 to
Eduard 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: :$\lim_f\left(x\right)=L$ if, and only if, for all sequences $x_n$ (with $x_n$ not equal to ''a'' for all ''n'') converging to $a$ the sequence $f\left(x_n\right)$ 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 $f\left(x_n\right)$ 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: :$\lim_f\left(x\right)=L$ if and only if for all $x\in \mathbb^*$, $f^*\left(x\right)-L$ is infinitesimal whenever $x-a$ is infinitesimal. Here $\mathbb^*$ are the
hyperreal 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
definition of limit 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 $A\subseteq \mathbb$ if for every $r>0$ there is a point $a\in A$ so that

# 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 be
continuous 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'': : $\lim_ f\left(x\right) = f\left(c\right).$ (We have here assumed that ''c'' is 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 the domain of ''f''.)

# 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'' is
continuous 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., $f+g$'','' $f-g$'','' $f\times g$'','' $f/g$'','' $f^g$) 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). :$\begin \lim\limits_ & \left(f\left(x\right) + g\left(x\right)\right) & = & \lim\limits_ f\left(x\right) + \lim\limits_ g\left(x\right) \\ \lim\limits_ & \left(f\left(x\right) - g\left(x\right)\right) & = & \lim\limits_ f\left(x\right) - \lim\limits_ g\left(x\right) \\ \lim\limits_ & \left(f\left(x\right)\cdot g\left(x\right)\right) & = & \lim\limits_ f\left(x\right) \cdot \lim\limits_ g\left(x\right) \\ \lim\limits_ & \left(f\left(x\right)/g\left(x\right)\right) & = & \\ \lim\limits_ & f\left(x\right)^ & = & \end$ 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'' ≠ −∞ *∞''q'' = 0 if ''q'' < 0 *∞''q'' = ∞ if ''q'' > 0 *''q'' = 0 if 0 < ''q'' < 1 *''q'' = ∞ if ''q'' > 1 *''q''−∞ = ∞ if 0 < ''q'' < 1 *''q''−∞ = 0 if ''q'' > 1 (see also
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 × ±∞ * ∞ + −∞ * 00 * ∞0 * 1±∞ See further
L'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 :$\lim_ f\left(y\right) = c$ and $\lim_ g\left(x\right) = b$, it does ''not'' follow that $\lim_ f\left(g\left(x\right)\right) = c$. 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 $\delta >0$ such that if $0<, x-a, <\delta$ then $, g\left(x\right)-b, >0$). As an example of this phenomenon, consider the following functions that violates both additional restrictions: :$f\left(x\right)=g\left(x\right)=\begin0 & \text x\neq 0 \\ 1 & \text x=0 \end.$ Since the value at ''f''(0) is a removable discontinuity, :$\lim_ f\left(x\right) = 0$ 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 :$f\left(f\left(x\right)\right)=\begin1 & \text x\neq 0 \\ 0 & \text x=0 \end$ and so :$\lim_ f\left(f\left(x\right)\right) = 1$ for all $a$.

## Limits of special interest

### Rational functions

For $n$ a nonnegative integer and constants $a_1, a_2, a_3,\ldots, a_n$ and $b_1, b_2, b_3,\ldots, b_n$, *$\lim_ \frac = \frac$ 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

*$\lim_ \frac = 1$ *$\lim_ \frac = 0$

### Exponential functions

*$\lim_ \left(1+x\right)^ = \lim_ \left\left(1+\frac\right\right)^ = e$ *$\lim_ \frac = 1$ *$\lim_ \frac = \frac$ *$\lim_ \frac = \frac\ln c$ *$\lim_ x^ = 1$

### Logarithmic functions

*$\lim_ \frac = 1$ *$\lim_ \frac = \frac$ *$\lim_ \frac = \frac$

## L'Hôpital's rule

This rule uses
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 ... 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: # $\lim_f\left(x\right)=\lim_g\left(x\right)=0,$ or $\lim_f\left(x\right)=\pm\lim_g\left(x\right) = \pm\infty$, and # $f$ and $g$ are differentiable over $I \setminus \$, and # $g\text{'}\left(x\right)\neq 0$ for all $x \in I \setminus \$, and # $\lim_\frac$ exists, then: $\lim_ \frac = \lim_ \frac$ Normally, the first condition is the most important one. For example: $\lim_ \frac = \lim_ \frac = \frac = \frac.$

## 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 $\lim_ \sum_^n f\left(i\right)$ is $\sum_^\infty f\left(i\right)$. An important example of limits of sums such as these are
series 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 $\lim_ \int_a^x f\left(t\right) \; dt$ is $\int_a^\infty f\left(t\right) \; dt$. A short way to write the limit $\lim_ \int_x^b f\left(t\right) \; dt$ is $\int_^b f\left(t\right) \; dt$.

* * * * * * * * *

# References

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