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Limit (mathematics)

From Wikipedia, the free encyclopedia
Value approached by a mathematical object
For other uses, seeLimit § Mathematics.

Inmathematics, alimit is thevalue that afunction (orsequence) approaches as theargument (or index) approaches some value.[1]Limits of functions are essential tocalculus andmathematical analysis, and are used to definecontinuity,derivatives, andintegrals.The concept of alimit of a sequence is further generalized to the concept of a limit of atopological net, and is closely related tolimit anddirect limit incategory theory.Thelimit inferior and limit superior provide generalizations of the concept of a limit which are particularly relevant when the limit at a point may not exist.

Notation

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In formulas, a limit of a function is usually written aslimx→cf(x)=L,{\displaystyle \lim _{x\to c}f(x)=L,}and is read as "the limit off ofx asx approachesc equalsL". This means that the value of the functionf can be made arbitrarily close toL, by choosingx sufficiently close toc. Alternatively, the fact that a functionf approaches the limitL asx approachesc is sometimes denoted by a right arrow (→ or→{\displaystyle \rightarrow }), as inf(x)→L as x→c,{\displaystyle f(x)\to L{\text{ as }}x\to c,}which reads "f{\displaystyle f} ofx{\displaystyle x} tends toL{\displaystyle L} asx{\displaystyle x} tends toc{\displaystyle c}".

History

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According toHankel (1871), the modern concept of limit originates from Proposition X.1 ofEuclid's Elements, which forms the basis of theMethod of exhaustion found in Euclid and Archimedes: "Two unequal magnitudes being set out, if from the greater there is subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process is repeated continually, then there will be left some magnitude less than the lesser magnitude set out."[2][3]

Grégoire de Saint-Vincent gave the first definition of limit (terminus) of ageometric series in his workOpus Geometricum (1647): "Theterminus of a progression is the end of the series, which none progression can reach, even not if she is continued in infinity, but which she can approach nearer than a given segment."[4]

In the Scholium toPrincipia in 1687,Isaac Newton had a clear definition of a limit, stating that "Those ultimate ratios ... are not actually ratios of ultimate quantities, but limits ... which they can approach so closely that their difference is less than any given quantity".[5]

The modern definition of a limit goes back toBernard Bolzano who, in 1817, developed the basics of theepsilon-delta technique to define continuous functions. However, his work remained unknown to other mathematicians until thirty years after his death.[6]

Augustin-Louis Cauchy in 1821,[7] followed byKarl Weierstrass, formalized the definition of the limit of a function which became known as the(ε, δ)-definition of limit.

The modern notation of placing the arrow below the limit symbol was invented by John Gaston Leathem in 1905 and popularized byG. H. Hardy's 1908 textbookA Course of Pure Mathematics.[8]

Types of limits

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In sequences

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Main article:Limit of a sequence

Real numbers

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The expression0.999... should be interpreted as the limit of the sequence 0.9, 0.99, 0.999, ... and so on. This sequence can be rigorously shown to have the limit 1, and therefore this expression is meaningfully interpreted as having the value 1.[9]

Formally, supposea1,a2, ... is asequence ofreal numbers. When the limit of the sequence exists, the real numberL is thelimit of this sequenceif and only if for everyreal numberε > 0, there exists anatural numberN such that for alln >N, we have|an −L| <ε.[10]The common notationlimn→∞an=L{\displaystyle \lim _{n\to \infty }a_{n}=L}is read as:

The limit ofan asn approaches infinity equalsL

or

The limit asn approaches infinity ofan equalsL.

The formal definition intuitively means that eventually, all elements of the sequence get arbitrarily close to the limit, since theabsolute value|an −L| is the distance betweenan andL.

Not every sequence has a limit. A sequence with a limit is calledconvergent; otherwise it is calleddivergent. 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 asn approaches infinity of a sequence{an} is simply the limit at infinity of a functiona(n)—defined on thenatural numbers{n}. On the other hand, ifX is the domain of a functionf(x) and if the limit asn approaches infinity off(xn) isL forevery arbitrary sequence of points{xn} inX −x0 which converges tox0, then the limit of the functionf(x) asx approachesx0 is equal toL.[11] One such sequence would be{x0 + 1/n}.

Infinity as a limit

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There is also a notion of having a limit "tend to infinity", rather than to a finite valueL{\displaystyle L}. A sequence{an}{\displaystyle \{a_{n}\}} is said to "tend to infinity" if, for each real numberM>0{\displaystyle M>0}, known as the bound, there exists an integerN{\displaystyle N} such that for eachn>N{\displaystyle n>N},an>M.{\displaystyle a_{n}>M.}That is, for every possible bound, the sequence eventually exceeds the bound. This is often writtenlimn→∞an=∞{\displaystyle \lim _{n\rightarrow \infty }a_{n}=\infty } or simplyan→∞{\displaystyle a_{n}\rightarrow \infty }.

It is possible for a sequence to be divergent, but not tend to infinity. Such sequences are calledoscillatory. An example of an oscillatory sequence isan=(−1)n{\displaystyle a_{n}=(-1)^{n}}.

There is a corresponding notion of tending to negative infinity,limn→∞an=−∞{\displaystyle \lim _{n\rightarrow \infty }a_{n}=-\infty }, defined by changing the inequality in the above definition toan<M,{\displaystyle a_{n}<M,} withM<0.{\displaystyle M<0.}

A sequence{an}{\displaystyle \{a_{n}\}} withlimn→∞|an|=∞{\displaystyle \lim _{n\rightarrow \infty }|a_{n}|=\infty } is calledunbounded, a definition equally valid for sequences in thecomplex numbers, or in anymetric space. Sequences which do not tend to infinity are calledbounded. Sequences which do not tend to positive infinity are calledbounded above, while those which do not tend to negative infinity arebounded below.

Metric space

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The discussion of sequences above is for sequences of real numbers. The notion of limits can be defined for sequences valued in more abstract spaces, such asmetric spaces. IfM{\displaystyle M} is a metric space with distance functiond{\displaystyle d}, and{an}n≥0{\displaystyle \{a_{n}\}_{n\geq 0}} is a sequence inM{\displaystyle M}, then the limit (when it exists) of the sequence is an elementa∈M{\displaystyle a\in M} such that, givenε>0{\displaystyle \varepsilon >0}, there exists anN{\displaystyle N} such that for eachn>N{\displaystyle n>N}, we haved(a,an)<ε.{\displaystyle d(a,a_{n})<\varepsilon .}An equivalent statement is thatan→a{\displaystyle a_{n}\rightarrow a} if the sequence of real numbersd(a,an)→0{\displaystyle d(a,a_{n})\rightarrow 0}.

Example: ℝn
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An important example is the space ofn{\displaystyle n}-dimensional real vectors, with elementsx=(x1,⋯,xn){\displaystyle \mathbf {x} =(x_{1},\cdots ,x_{n})} where each of thexi{\displaystyle x_{i}} are real, an example of a suitable distance function is theEuclidean distance, defined byd(x,y)=‖x−y‖=∑i(xi−yi)2.{\displaystyle d(\mathbf {x} ,\mathbf {y} )=\|\mathbf {x} -\mathbf {y} \|={\sqrt {\sum _{i}(x_{i}-y_{i})^{2}}}.}The sequence of points{xn}n≥0{\displaystyle \{\mathbf {x} _{n}\}_{n\geq 0}} converges tox{\displaystyle \mathbf {x} } if the limit exists and‖xn−x‖→0{\displaystyle \|\mathbf {x} _{n}-\mathbf {x} \|\rightarrow 0}.

Topological space

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In some sense themost abstract space in which limits can be defined aretopological spaces. IfX{\displaystyle X} is a topological space with topologyτ{\displaystyle \tau }, and{an}n≥0{\displaystyle \{a_{n}\}_{n\geq 0}} is a sequence inX{\displaystyle X}, then the limit (when it exists) of the sequence is a pointa∈X{\displaystyle a\in X} such that, given a (open)neighborhoodU∈τ{\displaystyle U\in \tau } ofa{\displaystyle a}, there exists anN{\displaystyle N} such that for everyn>N{\displaystyle n>N},an∈U{\displaystyle a_{n}\in U}is satisfied. In this case, the limit (if it exists) may not be unique. However it must be unique ifX{\displaystyle X} is aHausdorff space.

Function space

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This section deals with the idea of limits of sequences of functions, not to be confused with the idea of limits of functions, discussed below.

The field offunctional analysis partly seeks to identify useful notions of convergence on function spaces. For example, consider the space of functions from a generic setE{\displaystyle E} toR{\displaystyle \mathbb {R} }. Given a sequence of functions{fn}n>0{\displaystyle \{f_{n}\}_{n>0}} such that each is a functionfn:E→R{\displaystyle f_{n}:E\rightarrow \mathbb {R} }, suppose that there exists a function such that for each {{nowrap|x∈E{\displaystyle x\in E},{{lang|la|fn(x)→f(x) or equivalently limn→∞fn(x)=f(x).{\displaystyle f_{n}(x)\rightarrow f(x){\text{ or equivalently }}\lim _{n\rightarrow \infty }f_{n}(x)=f(x).}

Then the sequencefn{\displaystyle f_{n}} is said toconverge pointwise tof{\displaystyle f}. However, such sequences can exhibit unexpected behavior. For example, it is possible to construct a sequence of continuous functions which has a discontinuous pointwise limit.

Another notion of convergence isuniform convergence. The uniform distance between two functionsf,g:E→R{\displaystyle f,g:E\rightarrow \mathbb {R} } is the maximum difference between the two functions as the argumentx∈E{\displaystyle x\in E} is varied. That is,d(f,g)=maxx∈E|f(x)−g(x)|.{\displaystyle d(f,g)=\max _{x\in E}|f(x)-g(x)|.}Then the sequencefn{\displaystyle f_{n}} is said touniformly converge or have auniform limit off{\displaystyle f} iffn→f{\displaystyle f_{n}\rightarrow f} with respect to this distance. The uniform limit has "nicer" properties than the pointwise limit. For example, the uniform limit of a sequence of continuous functions is continuous.

Many different notions of convergence can be defined on function spaces. This is sometimes dependent on theregularity of the space. Prominent examples of function spaces with some notion of convergence areLp spaces andSobolev space.

In functions

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Main article:Limit of a function
A functionf(x) for which thelimit at infinity isL. For any arbitrary distanceε, there must be a valueS such that the function stays withinL ±ε for allx >S.

Supposef is areal-valued function andc is areal number. Intuitively speaking, the expression

limx→cf(x)=L{\displaystyle \lim _{x\to c}f(x)=L}

means thatf(x) can be made to be as close toL as desired, by makingx sufficiently close toc.[12] In that case, the above equation can be read as "the limit off ofx, asx approachesc, isL".

Formally, the definition of the "limit off(x){\displaystyle f(x)} asx{\displaystyle x} approachesc{\displaystyle c}" is given as follows. The limit is a real numberL{\displaystyle L} so that, given an arbitrary real numberε>0{\displaystyle \varepsilon >0} (thought of as the "error"), there is aδ>0{\displaystyle \delta >0} such that, for anyx{\displaystyle x} satisfying0<|x−c|<δ{\displaystyle 0<|x-c|<\delta }, it holds that|f(x)−L|<ε{\displaystyle |f(x)-L|<\varepsilon }. This is known as the(ε,δ)-definition of limit.

The inequality0<|x−c|{\displaystyle 0<|x-c|} is used to excludec{\displaystyle c} from the set of points under consideration, but some authors do not include this in their definition of limits, replacing0<|x−c|<δ{\displaystyle 0<|x-c|<\delta } with simply|x−c|<δ{\displaystyle |x-c|<\delta }. This replacement is equivalent to additionally requiring thatf{\displaystyle f} be continuous atc{\displaystyle c}.

It can be proven that there is an equivalent definition which makes manifest the connection between limits of sequences and limits of functions.[13] The equivalent definition is given as follows. First observe that for every sequence{xn}{\displaystyle \{x_{n}\}} in the domain off{\displaystyle f}, there is an associated sequence{f(xn)}{\displaystyle \{f(x_{n})\}}, the image of the sequence underf{\displaystyle f}. The limit is a real numberL{\displaystyle L} so that, forall sequencesxn→c{\displaystyle x_{n}\rightarrow c}, the associated sequencef(xn)→L{\displaystyle f(x_{n})\rightarrow L}.

One-sided limit

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Main article:One-sided limit

It is possible to define the notion of having a "left-handed" limit ("from below"), and a notion of a "right-handed" limit ("from above"). These need not agree. An example is given by the positiveindicator function,f:R→R{\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }, defined such thatf(x)=0{\displaystyle f(x)=0} ifx≤0{\displaystyle x\leq 0}, andf(x)=1{\displaystyle f(x)=1} ifx>0{\displaystyle x>0}. Atx=0{\displaystyle x=0}, the function has a "left-handed limit" of 0, a "right-handed limit" of 1, and its limit does not exist. Symbolically, this can be stated as, for this example,limx→c−f(x)=0{\displaystyle \lim _{x\to c^{-}}f(x)=0}, andlimx→c+f(x)=1{\displaystyle \lim _{x\to c^{+}}f(x)=1}, and from this it can be deducedlimx→cf(x){\displaystyle \lim _{x\to c}f(x)} does not exist, becauselimx→c−f(x)≠limx→c+f(x){\displaystyle \lim _{x\to c^{-}}f(x)\neq \lim _{x\to c^{+}}f(x)}.

Infinity in limits of functions

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It is possible to define the notion of "tending to infinity" in the domain off{\displaystyle f},limx→+∞f(x)=L.{\displaystyle \lim _{x\rightarrow +\infty }f(x)=L.}

This could be considered equivalent to the limit as a reciprocal tends to 0:limx′→0+f(1/x′)=L.{\displaystyle \lim _{x'\rightarrow 0^{+}}f(1/x')=L.}

or it can be defined directly: the "limit off{\displaystyle f} asx{\displaystyle x} tends to positive infinity" is defined as a valueL{\displaystyle L} such that, given any realε>0{\displaystyle \varepsilon >0}, there exists anM>0{\displaystyle M>0} so that for allx>M{\displaystyle x>M},|f(x)−L|<ε{\displaystyle |f(x)-L|<\varepsilon }. The definition for sequences is equivalent: Asn→+∞{\displaystyle n\rightarrow +\infty }, we havef(xn)→L{\displaystyle f(x_{n})\rightarrow L}.

In these expressions, the infinity is normally considered to be signed (+∞{\displaystyle +\infty } or−∞{\displaystyle -\infty }) and corresponds to a one-sided limit of the reciprocal. A two-sided infinite limit can be defined, but an author would explicitly write±∞{\displaystyle \pm \infty } to be clear.

It is also possible to define the notion of "tending to infinity" in the value off{\displaystyle f},limx→cf(x)=∞.{\displaystyle \lim _{x\rightarrow c}f(x)=\infty .}

Again, this could be defined in terms of a reciprocal:limx→c1f(x)=0.{\displaystyle \lim _{x\rightarrow c}{\frac {1}{f(x)}}=0.}

Or a direct definition can be given as follows: given any real numberM>0{\displaystyle M>0}, there is aδ>0{\displaystyle \delta >0} so that for0<|x−c|<δ{\displaystyle 0<|x-c|<\delta }, the absolute value of the function|f(x)|>M{\displaystyle |f(x)|>M}. A sequence can also have an infinite limit: asn→∞{\displaystyle n\rightarrow \infty }, the sequencef(xn)→∞{\displaystyle f(x_{n})\rightarrow \infty }.

This direct definition is easier to extend to one-sided infinite limits. While mathematicians do talk about functions approaching limits "from above" or "from below", there is not a standardmathematical notation for this as there is for one-sided limits.

Nonstandard analysis

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Innon-standard analysis (which involves ahyperreal enlargement of the number system), the limit of a sequence(an){\displaystyle (a_{n})} can be expressed as thestandard part of the valueaH{\displaystyle a_{H}} of the natural extension of the sequence at an infinitehypernatural indexn =H. Thus,limn→∞an=st⁡(aH).{\displaystyle \lim _{n\to \infty }a_{n}=\operatorname {st} (a_{H}).}Here, the standard part function "st" rounds off each finite hyperreal number to the nearest real number (the difference between them isinfinitesimal). 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 hyperreala=[an]{\displaystyle a=[a_{n}]} represented in the ultrapower construction by a Cauchy sequence(an){\displaystyle (a_{n})}, is simply the limit of that sequence:st⁡(a)=limn→∞an.{\displaystyle \operatorname {st} (a)=\lim _{n\to \infty }a_{n}.}In this sense, taking the limit and taking the standard part are equivalent procedures.

Limit sets

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Limit set of a sequence

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Let{an}n>0{\displaystyle \{a_{n}\}_{n>0}} be a sequence in a topological spaceX{\displaystyle X}. For concreteness,X{\displaystyle X} can be thought of asR{\displaystyle \mathbb {R} }, but the definitions hold more generally. Thelimit set is the set of points such that if there is a convergentsubsequence{ank}k>0{\displaystyle \{a_{n_{k}}\}_{k>0}} withank→a{\displaystyle a_{n_{k}}\rightarrow a}, thena{\displaystyle a} belongs to the limit set. In this context, such ana{\displaystyle a} is sometimes called a limit point.

A use of this notion is to characterize the "long-term behavior" of oscillatory sequences. For example, consider the sequencean=(−1)n{\displaystyle a_{n}=(-1)^{n}}. Starting from n=1, the first few terms of this sequence are−1,+1,−1,+1,⋯{\displaystyle -1,+1,-1,+1,\cdots }. It can be checked that it is oscillatory, so has no limit, but has limit points{−1,+1}{\displaystyle \{-1,+1\}}.

Limit set of a trajectory

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This notion is used indynamical systems, to study limits of trajectories. Defining a trajectory to be a functionγ:R→X{\displaystyle \gamma :\mathbb {R} \rightarrow X}, the pointγ(t){\displaystyle \gamma (t)} is thought of as the "position" of the trajectory at "time"t{\displaystyle t}. The limit set of a trajectory is defined as follows. To any sequence of increasing times{tn}{\displaystyle \{t_{n}\}}, there is an associated sequence of positions{xn}={γ(tn)}{\displaystyle \{x_{n}\}=\{\gamma (t_{n})\}}. Ifx{\displaystyle x} is the limit set of the sequence{xn}{\displaystyle \{x_{n}\}} for any sequence of increasing times, thenx{\displaystyle x} is a limit set of the trajectory.

Technically, this is theω{\displaystyle \omega }-limit set. The corresponding limit set for sequences of decreasing time is called theα{\displaystyle \alpha }-limit set.

An illustrative example is the circle trajectory:γ(t)=(cos⁡(t),sin⁡(t)){\displaystyle \gamma (t)=(\cos(t),\sin(t))}. This has no unique limit, but for eachθ∈R{\displaystyle \theta \in \mathbb {R} }, the point(cos⁡(θ),sin⁡(θ)){\displaystyle (\cos(\theta ),\sin(\theta ))} is a limit point, given by the sequence of timestn=θ+2πn{\displaystyle t_{n}=\theta +2\pi n}. But the limit points need not be attained on the trajectory. The trajectoryγ(t)=t/(1+t)(cos⁡(t),sin⁡(t)){\displaystyle \gamma (t)=t/(1+t)(\cos(t),\sin(t))} also has theunit circle as its limit set.

Uses

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Limits are used to define a number of important concepts in analysis.

Series

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Main article:series (mathematics)

A particular expression of interest which is formalized as the limit of a sequence is sums of infinite series. These are "infinite sums" of real numbers, generally written as∑n=1∞an.{\displaystyle \sum _{n=1}^{\infty }a_{n}.}This is defined through limits as follows:[13] given a sequence of real numbers{an}{\displaystyle \{a_{n}\}}, the sequence of partial sums is defined bysn=∑i=1nai.{\displaystyle s_{n}=\sum _{i=1}^{n}a_{i}.}If the limit of the sequence{sn}{\displaystyle \{s_{n}\}} exists, the value of the expression∑n=1∞an{\displaystyle \sum _{n=1}^{\infty }a_{n}} is defined to be the limit. Otherwise, the series is said to be divergent.

A classic example is theBasel problem, wherean=1/n2{\displaystyle a_{n}=1/n^{2}}. Then∑n=1∞1n2=π26.{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}.}

However, while for sequences there is essentially a unique notion of convergence, for series there are different notions of convergence. This is due to the fact that the expression∑n=1∞an{\displaystyle \sum _{n=1}^{\infty }a_{n}} does not discriminate between different orderings of the sequence{an}{\displaystyle \{a_{n}\}}, while the convergence properties of the sequence of partial sumscan depend on the ordering of the sequence.

A series which converges for all orderings is calledunconditionally convergent. It can be proven to be equivalent toabsolute convergence. This is defined as follows. A series is absolutely convergent if∑n=1∞|an|{\displaystyle \sum _{n=1}^{\infty }|a_{n}|} is well defined. Furthermore, all possible orderings give the same value.

Otherwise, the series isconditionally convergent. A surprising result for conditionally convergent series is theRiemann series theorem: depending on the ordering, the partial sums can be made to converge to any real number, as well as±∞{\displaystyle \pm \infty }.

Power series

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Main article:Power series

A useful application of the theory of sums of series is forpower series. These are sums of series of the formf(z)=∑n=0∞cnzn.{\displaystyle f(z)=\sum _{n=0}^{\infty }c_{n}z^{n}.}Oftenz{\displaystyle z} is thought of as a complex number, and a suitable notion of convergence of complex sequences is needed. The set of values ofz∈C{\displaystyle z\in \mathbb {C} } for which the series sum converges is a circle, with its radius known as theradius of convergence.

Continuity of a function at a point

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The definition of continuity at a point is given through limits.

The above definition of a limit is true even iff(c)≠L{\displaystyle f(c)\neq L}. Indeed, the functionf need not even be defined atc. However, iff(c){\displaystyle f(c)} is defined and is equal toL{\displaystyle L}, then the function is said to becontinuous at the pointc{\displaystyle c}.

Equivalently, the function is continuous atc{\displaystyle c} iff(x)→f(c){\displaystyle f(x)\rightarrow f(c)} asx→c{\displaystyle x\rightarrow c}, or in terms of sequences, wheneverxn→c{\displaystyle x_{n}\rightarrow c}, thenf(xn)→f(c){\displaystyle f(x_{n})\rightarrow f(c)}.

An example of a limit wheref{\displaystyle f} is not defined atc{\displaystyle c} is given below.

Consider the function

f(x)=x2−1x−1.{\displaystyle f(x)={\frac {x^{2}-1}{x-1}}.}

thenf(1) is not defined (seeIndeterminate form), yet asx moves arbitrarily close to 1,f(x) correspondingly approaches 2:[14]

f(0.9)f(0.99)f(0.999)f(1.0)f(1.001)f(1.01)f(1.1)
1.9001.9901.999undefined2.0012.0102.100

Thus,f(x) can be made arbitrarily close to the limit of 2—just by makingx sufficiently close to1. In other words,limx→1x2−1x−1=2.{\displaystyle \lim _{x\to 1}{\frac {x^{2}-1}{x-1}}=2.}

This can also be calculated algebraically, asx2−1x−1=(x+1)(x−1)x−1=x+1{\textstyle {\frac {x^{2}-1}{x-1}}={\frac {(x+1)(x-1)}{x-1}}=x+1} for all real numbersx ≠ 1.

Now, sincex + 1 is continuous inx at 1, we can now plug in 1 forx, leading to the equationlimx→1x2−1x−1=1+1=2.{\displaystyle \lim _{x\to 1}{\frac {x^{2}-1}{x-1}}=1+1=2.}

In addition to limits at finite values, functions can also have limits at infinity. For example, consider the functionf(x)=2x−1x{\displaystyle f(x)={\frac {2x-1}{x}}}where:

  • f(100) = 1.9900
  • f(1000) = 1.9990
  • f(10000) = 1.9999

Asx becomes extremely large, the value off(x) approaches2, and the value off(x) can be made as close to2 as one could wish—by makingx sufficiently large. So in this case, the limit off(x) asx approaches infinity is2, or in mathematical notation,limx→∞2x−1x=2.{\displaystyle \lim _{x\to \infty }{\frac {2x-1}{x}}=2.}

Continuous functions

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An important class of functions when considering limits arecontinuous functions. These are precisely those functions whichpreserve limits, in the sense that iff{\displaystyle f} is a continuous function, then wheneveran→a{\displaystyle a_{n}\rightarrow a} in the domain off{\displaystyle f}, then the limitf(an){\displaystyle f(a_{n})} exists and furthermore isf(a){\displaystyle f(a)}.

In the most general setting of topological spaces, a short proof is given below:

Letf:X→Y{\displaystyle f:X\rightarrow Y} be a continuous function between topological spacesX{\displaystyle X} andY{\displaystyle Y}. By definition, for each open setV{\displaystyle V} inY{\displaystyle Y}, the preimagef−1(V){\displaystyle f^{-1}(V)} is open inX{\displaystyle X}.

Now supposean→a{\displaystyle a_{n}\rightarrow a} is a sequence with limita{\displaystyle a} inX{\displaystyle X}. Thenf(an){\displaystyle f(a_{n})} is a sequence inY{\displaystyle Y}, andf(a){\displaystyle f(a)} is some point.

Choose a neighborhoodV{\displaystyle V} off(a){\displaystyle f(a)}. Thenf−1(V){\displaystyle f^{-1}(V)} is anopen set (by continuity off{\displaystyle f}) which in particular containsa{\displaystyle a}, and thereforef−1(V){\displaystyle f^{-1}(V)} is a neighborhood ofa{\displaystyle a}. By the convergence ofan{\displaystyle a_{n}} toa{\displaystyle a}, there exists anN{\displaystyle N} such that forn>N{\displaystyle n>N}, we havean∈f−1(V){\displaystyle a_{n}\in f^{-1}(V)}.

Then applyingf{\displaystyle f} to both sides gives that, for the sameN{\displaystyle N}, for eachn>N{\displaystyle n>N} we havef(an)∈V{\displaystyle f(a_{n})\in V}. OriginallyV{\displaystyle V} was an arbitrary neighborhood off(a){\displaystyle f(a)}, sof(an)→f(a){\displaystyle f(a_{n})\rightarrow f(a)}. This concludes the proof.

In real analysis, for the more concrete case of real-valued functions defined on a subsetE⊂R{\displaystyle E\subset \mathbb {R} }, that is,f:E→R{\displaystyle f:E\rightarrow \mathbb {R} }, a continuous function may also be defined as a function which is continuous at every point of its domain.

Limit points

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Intopology, limits are used to definelimit points of a subset of a topological space, which in turn give a useful characterization ofclosed sets.

In a topological spaceX{\displaystyle X}, consider a subsetS{\displaystyle S}. A pointa{\displaystyle a} is called a limit point if there is a sequence{an}{\displaystyle \{a_{n}\}} inS∖{a}{\displaystyle S\setminus \{a\}} such thatan→a{\displaystyle a_{n}\rightarrow a}.

The reason why{an}{\displaystyle \{a_{n}\}} is defined to be inS∖{a}{\displaystyle S\setminus \{a\}} rather than justS{\displaystyle S} is illustrated by the following example. TakeX=R{\displaystyle X=\mathbb {R} } andS=[0,1]∪{2}{\displaystyle S=[0,1]\cup \{2\}}. Then2∈S{\displaystyle 2\in S}, and therefore is the limit of the constant sequence2,2,⋯{\displaystyle 2,2,\cdots }. But2{\displaystyle 2} is not a limit point ofS{\displaystyle S}.

A closed set, which is defined to be the complement of an open set, is equivalently any setC{\displaystyle C} which contains all its limit points.

Derivative

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Main article:derivative

The derivative is defined formally as a limit. In the scope ofreal analysis, the derivative is first defined for real functionsf{\displaystyle f} defined on a subsetE⊂R{\displaystyle E\subset \mathbb {R} }. The derivative atx∈E{\displaystyle x\in E} is defined as follows. If the limit off(x+h)−f(x)h{\displaystyle {\frac {f(x+h)-f(x)}{h}}}ash→0{\displaystyle h\rightarrow 0} exists, then the derivative atx{\displaystyle x} is this limit.

Equivalently, it is the limit asy→x{\displaystyle y\rightarrow x} off(y)−f(x)y−x.{\displaystyle {\frac {f(y)-f(x)}{y-x}}.}

If the derivative exists, it is commonly denoted byf′(x){\displaystyle f'(x)}.

Properties

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Sequences of real numbers

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For sequences of real numbers, a number of properties can be proven.[13] Suppose{an}{\displaystyle \{a_{n}\}} and{bn}{\displaystyle \{b_{n}\}} are two sequences converging toa{\displaystyle a} andb{\displaystyle b} respectively.

Equivalently, the functionf(x)=1/x{\displaystyle f(x)=1/x} is continuous about nonzerox{\displaystyle x}.

Cauchy sequences

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See also:Cauchy sequence

A property of convergent sequences of real numbers is that they areCauchy sequences.[13] The definition of a Cauchy sequence{an}{\displaystyle \{a_{n}\}} is that for every real numberε>0{\displaystyle \varepsilon >0}, there is anN{\displaystyle N} such that wheneverm,n>N{\displaystyle m,n>N},|am−an|<ε.{\displaystyle |a_{m}-a_{n}|<\varepsilon .}

Informally, for any arbitrarily small errorε{\displaystyle \varepsilon }, it is possible to find an interval of diameterε{\displaystyle \varepsilon } such that eventually the sequence is contained within the interval.

Cauchy sequences are closely related to convergent sequences. In fact, for sequences of real numbers they are equivalent: any Cauchy sequence is convergent.

In general metric spaces, it continues to hold that convergent sequences are also Cauchy. But the converse is not true: not every Cauchy sequence is convergent in a general metric space. A classic counterexample is therational numbers,Q{\displaystyle \mathbb {Q} }, with the usual distance. The sequence of decimal approximations to2{\displaystyle {\sqrt {2}}}, truncated at then{\displaystyle n}th decimal place is a Cauchy sequence, butdoes not converge inQ{\displaystyle \mathbb {Q} }.

A metric space in which every Cauchy sequence is also convergent, that is, Cauchy sequences are equivalent to convergent sequences, is known as acomplete metric space.

One reason Cauchy sequences can be "easier to work with" than convergent sequences is that they are a property of the sequence{an}{\displaystyle \{a_{n}\}} alone, while convergent sequences require not just the sequence{an}{\displaystyle \{a_{n}\}} but also the limit of the sequencea{\displaystyle a}.

Order of convergence

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Beyond whether or not a sequence{an}{\displaystyle \{a_{n}\}} converges to a limita{\displaystyle a}, it is possible to describe how fast a sequence converges to a limit. One way to quantify this is using theorder of convergence of a sequence.

A formal definition of order of convergence can be stated as follows. Suppose{an}n>0{\displaystyle \{a_{n}\}_{n>0}} is a sequence of real numbers which is convergent with limita{\displaystyle a}. Furthermore,an≠a{\displaystyle a_{n}\neq a} for alln{\displaystyle n}. If positive constantsλ{\displaystyle \lambda } andα{\displaystyle \alpha } exist such thatlimn→∞|an+1−a||an−a|α=λ{\displaystyle \lim _{n\to \infty }{\frac {\left|a_{n+1}-a\right|}{\left|a_{n}-a\right|^{\alpha }}}=\lambda }thenan{\displaystyle a_{n}} is said to converge toa{\displaystyle a} withorder of convergenceα{\displaystyle \alpha }. The constantλ{\displaystyle \lambda } is known as the asymptotic error constant.

Order of convergence is used for example the field ofnumerical analysis, in error analysis.

Computability

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Limits can be difficult to compute. There exist limit expressions whosemodulus of convergence isundecidable. Inrecursion theory, thelimit lemma proves that it is possible to encode undecidable problems using limits.[15]

There are several theorems or tests that indicate whether the limit exists. These are known asconvergence tests. Examples include theratio test and thesqueeze theorem. However they may not tell how to compute the limit.

See also

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Notes

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  1. ^Stewart, James (2008).Calculus: Early Transcendentals (6th ed.).Brooks/Cole.ISBN 978-0-495-01166-8.
  2. ^Schubring, Gert (2005).Conflicts between generalization, rigor, and intuition: number concepts underlying the development of analysis in 17th–19th century France and Germany. New York: Springer. pp. 22–23.ISBN 0387228365.
  3. ^Euclid.Elements. Translated by Joyce, David E. Worcester, Massachusetts: Clark University. Book X, Proposition 1.
  4. ^Van Looy, Herman (1984)."A chronology and historical analysis of the mathematical manuscripts of Gregorius a Sancto Vincentio (1584–1667)".Historia Mathematica.11 (1):57–75.doi:10.1016/0315-0860(84)90005-3.
  5. ^Rowlands, Peter (2017).Newton and the Great World System.World Scientific. p. 28.doi:10.1142/q0108.ISBN 978-1-78634-372-7.
  6. ^Felscher, Walter (2000). "Bolzano, Cauchy, Epsilon, Delta".American Mathematical Monthly.107 (9):844–862.doi:10.2307/2695743.JSTOR 2695743.
  7. ^Larson, Ron; Edwards, Bruce H. (2010).Calculus of a single variable (9th ed.).Brooks/Cole,Cengage Learning.ISBN 978-0-547-20998-2.
  8. ^Miller, Jeff (1 December 2004). Robertson, Edmund; O'Connor, John (eds.)."Earliest Uses of Symbols of Calculus".MacTutor.Archived from the original on 2025-09-21. Retrieved2008-12-18.
  9. ^Stillwell, John (1994).Elements of algebra: geometry, numbers, equations. Springer. p. 42.ISBN 978-1441928399.
  10. ^Weisstein, Eric W."Limit".Wolfram MathWorld.Archived from the original on 2020-06-20. Retrieved2020-08-18.
  11. ^Apostol (1974, pp. 75–76)
  12. ^Weisstein, Eric W."Epsilon-Delta Definition".Wolfram MathWorld.Archived from the original on 2020-06-25. Retrieved2020-08-18.
  13. ^abcdGowers, Timothy; Chua, Dexter."Analysis I".Notes from the Mathematical Tripos.
  14. ^"limit".Encyclopædia Britannica.Archived from the original on 2021-05-09. Retrieved2020-08-18.
  15. ^Soare, Robert I. (2014).Recursively enumerable sets and degrees : a study of computable functions and computably generated sets. Berlin: Springer.ISBN 978-3-540-66681-3.OCLC 1154894968.

References

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External links

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