| 1 | 0.841471... |
| 0.1 | 0.998334... |
| 0.01 | 0.999983... |
Although the function is not defined at zero, asx becomes closer and closer to zero, becomes arbitrarily close to 1. In other words, the limit of asx approaches zero, equals 1.
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Inmathematics, thelimit of a function is a fundamental concept incalculus andanalysis concerning the behavior of thatfunction near a particularinput which may or may not be in thedomain of the function.
Formal definitions, first devised in the early 19th century, are given below. Informally, a functionf assigns anoutputf(x) to every inputx. We say that the function has a limitL at an inputp, iff(x) gets closer and closer toL asx moves closer and closer top. More specifically, the output value can be madearbitrarily close toL if the input tof is takensufficiently close top. On the other hand, if some inputs very close top are taken to outputs that stay a fixed distance apart, then we say the limitdoes not exist.
The notion of a limit has many applications inmodern calculus. In particular, the many definitions ofcontinuity 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 thederivative: in the calculus of one variable, this is the limiting value of theslope ofsecant lines to thegraph of a function.
Although implicit in thedevelopment of calculus of the 17th and 18th centuries, the modern idea of the limit of a function goes back toBernard Bolzano who, in 1817, introduced the basics of the epsilon-delta technique (see(ε, δ)-definition of limit below) to define continuous functions. However, his work was not known during his lifetime.[1] Bruce Pourciau argues thatIsaac Newton, in his 1687Principia, demonstrates a more sophisticated understanding of limits than he is generally given credit for, including being the first to present an epsilon argument.[2][3]
In his 1821 bookCours d'analyse,Augustin-Louis Cauchy discussed variable quantities,infinitesimals and limits, and defined continuity of by saying that an infinitesimal change inx necessarily produces an infinitesimal change iny, while Grabiner claims that he used a rigorous epsilon-delta definition in proofs.[4] In 1861,Karl Weierstrass first introduced the epsilon-delta definition of limit in the form it is usually written today.[5] He also introduced the notations and[6]
The modern notation of placing the arrow below the limit symbol is due toG. H. Hardy, which is introduced in his bookA Course of Pure Mathematics in 1908.[7]
Imagine a person walking on a landscape represented by the graphy =f(x). Their horizontal position is given byx, much like the position given by a map of the land or by aglobal positioning system. Their altitude is given by the coordinatey. Suppose they walk towards a positionx =p, as they get closer and closer to this point, they will notice that their altitude approaches a specific valueL. If asked about the altitude corresponding tox =p, they would reply by sayingy =L.
What, then, does it mean to say, their altitude is approachingL? It means that their altitude gets nearer and nearer toL—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 ofL. They report back that indeed, they can get within ten vertical meters ofL, arguing that as long as they are within fifty horizontal meters ofp, their altitude isalways within ten meters ofL.
The accuracy goal is then changed: can they get within one vertical meter? Yes, supposing that they are able to move within five horizontal meters ofp, their altitude will always remain within one meter from the target altitudeL. Summarizing the aforementioned concept we can say that the traveler's altitudef(x) approachesL as their horizontal positionx approachesp, so as to say that for every target accuracy goal, however small it may be, there is some neighbourhood ofp, within which for every memberx' the target accuracy goal is fulfilled by an altidudef'(x) , except for maybe the horizontal positionp itself.
The initial informal statement can now be explicated:
In fact, this explicit statement is quite close to the formal definition of the limit of a function, with values in atopological space.
More specifically, to say thatis to say thatf(x) can be made as close toL as desired, by makingx close enough, but not equal, to p.[8]
The following definitions, known as(ε,δ)-definitions, are the generally accepted definitions for the limit of a function in various contexts.
Suppose is a function defined on thereal line, and there are two real numbersp andL. One would say: "The limit off ofx, asx approachesp, exists, and it equalsL". and write,[9]or alternatively, say "f(x) tends toL asx tends top", and write,if the following property holds: for every realε > 0, there exists a realδ > 0 such that for all realx, 0 < |x −p| <δ implies|f(x) −L| <ε.[9] Symbolically,
For example, one may saybecause for every realε > 0, we can takeδ =ε/4, so that for all realx, if0 < |x − 2| <δ, then|4x + 1 − 9| <ε.
A more general definition applies for functions defined onsubsets of the real line. LetS be a subset of Let be areal-valued function. Letp be a point such that there exists some open interval(a,b) containingp with It is then said that the limit off asx approachesp isL, if:
Symbolically,
For example, one may saybecause for every realε > 0, we can takeδ =ε, so that for all realx ≥ −3, if0 < |x − 1| <δ, then|f(x) − 2| <ε. In this example,S = [−3, ∞) contains open intervals around the point 1 (for example, the interval (0, 2)).
Here, note that the value of the limit does not depend onf being defined atp, nor on the valuef(p)—if it is defined. For example, letbecause for everyε > 0, we can takeδ =ε/2, so that for all realx ≠ 1, if0 < |x − 1| <δ, then|f(x) − 3| <ε. Note that heref(1) is undefined.
In fact, a limit can exist in which equals whereintS is theinterior ofS, andisoSc are theisolated points of the complement ofS. In our previous example where We see, specifically, this definition of limit allows a limit to exist at 1, but not at 0 or 2.
The lettersε andδ can be understood as "error" and "distance". In fact, Cauchy usedε as an abbreviation for "error" in some of his work,[4] though in his definition of continuity, he used an infinitesimal rather than eitherε orδ (seeCours d'Analyse). 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 thatδ andε represent distances helps suggest these generalizations.
Alternatively,x may approachp from above (right) or below (left), in which case the limits may be written as
or
respectively. If these limits exist at p and are equal there, then this can be referred to asthe limit off(x) atp.[10] If the one-sided limits exist atp, but are unequal, then there is no limit atp (i.e., the limit atp does not exist). If either one-sided limit does not exist atp, then the limit atp also does not exist.
A formal definition is as follows. Thelimit off asx approachesp from above isL if:
Thelimit off asx approachesp from below isL if:
If the limit does not exist, then theoscillation off atp is non-zero.
Limits can also be defined by approaching from subsets of the domain.
In general:[11] Let be a real-valued function defined on some Letp be alimit point of some—that is,p is the limit of some sequence of elements ofT distinct fromp. Then we saythe limit off, asx approachesp from values inT, isL, writtenif the following holds:
Note,T can be any subset ofS, the domain off. And the limit might depend on the selection ofT. This generalization includes as special cases limits on an interval, as well as left-handed limits of real-valued functions (e.g., by takingT to be an open interval of the form(–∞,a)), and right-handed limits (e.g., by takingT to be an open interval of the form(a, ∞)). It also extends the notion of one-sided limits to the included endpoints of (half-)closed intervals, so thesquare root function can have limit 0 asx approaches 0 from above:since for everyε > 0, we may takeδ =ε2 such that for allx ≥ 0, if0 < |x − 0| <δ, then|f(x) − 0| <ε.
This definition allows a limit to be defined at limit points of the domainS, if a suitable subsetT which has the same limit point is chosen.
Notably, the previous two-sided definition works on which is a subset of the limit points ofS.
For example, let The previous two-sided definition would work at but it wouldn't work at 0 or 2, which are limit points ofS.
The definition of limit given here does not depend on how (or whether)f is defined atp. Bartle[12] refers to this as adeleted limit, because it excludes the value off atp. The correspondingnon-deleted limit does depend on the value off atp, ifp is in the domain off. Let be a real-valued function.The non-deleted limit off, asx approachesp, isL if
The definition is the same, except that the neighborhood|x −p| <δ now includes the pointp, in contrast to thedeleted neighborhood0 < |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 thetheorem about limits of compositions without any constraints on the functions (other than the existence of their non-deleted limits).[13]
Bartle[12] notes that although by "limit" some authors do mean this non-deleted limit, deleted limits are the most popular.[14]
The functionhas no limit atx0 = 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, see picture), but has a limit at every otherx-coordinate.
The function(a.k.a., theDirichlet function) has no limit at anyx-coordinate.
The functionhas a limit at every non-zerox-coordinate (the limit equals 1 for negativex and equals 2 for positivex). The limit atx = 0 does not exist (the left-hand limit equals 1, whereas the right-hand limit equals 2).
The functionsandboth have a limit atx = 0 and it equals 0.
The functionhas a limit at anyx-coordinate of the form wheren is any integer.
Let be a function defined onThe limit off asx approaches infinity isL, denoted
means that:
Similarly,the limit off asx approaches minus infinity isL, denoted
means that:
For example,because for everyε > 0, we can takec = 3/ε such that for all realx, ifx >c, then|f(x) − 4| <ε.
Another example is thatbecause for everyε > 0, we can takec = max{1, −ln(ε)} such that for all realx, ifx < −c, then|f(x) − 0| <ε.
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 be a function defined on The statementthe limit off asx approachesp is infinity, denoted
means that:
The statementthe limit off asx approachesp is minus infinity, denoted
means that:
For example,because for everyN > 0, we can take such that for all realx > 0, if0 <x − 1 <δ, thenf(x) >N.
These ideas can be used together to produce definitions for different combinations, such as
or
For example,because for everyN > 0, we can takeδ =e−N such that for all realx > 0, if0 <x − 0 <δ, thenf(x) < −N.
Limits involving infinity are connected with the concept ofasymptotes.
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
In this case, is a topological space and any function of the form with 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.
Many authors[15] allow for theprojectively extended real line to be used as a way to include infinite values as well asextended real line. With this notation, the extended real line is given as and the projectively extended real line is 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, does not possess a central limit (which is normal):
In contrast, when working with the projective real line, infinities (much like 0) are unsigned, so, the central limitdoes exist in that context:
In fact there are a plethora of conflicting formal systems in use.In certain applications ofnumerical differentiation and integration, it is, for example, convenient to havesigned zeroes. A simple reason has to do with the converse of namely, it is convenient for to be considered true.Such zeroes can be seen as an approximation toinfinitesimals.
There are three basic rules for evaluating limits at infinity for arational function (wherep andq are polynomials):
If the limit at infinity exists, it represents a horizontal asymptote aty =L. Polynomials do not have horizontal asymptotes; such asymptotes may however occur with rational functions.
By noting that|x −p| represents adistance, the definition of a limit can be extended to functions of more than one variable. In the case of a function defined on we defined the limit as follows:the limit off as(x,y) approaches(p,q) isL, written
if the following condition holds:
or formally:
Here is theEuclidean distance between(x,y) and(p,q). (This can in fact be replaced by anynorm||(x,y) − (p,q)||, and be extended to any number of variables.)
For example, we may saybecause for everyε > 0, we can take such that for all realx ≠ 0 and realy ≠ 0, if then|f(x,y) − 0| <ε.
Similar to the case in single variable, the value off at(p,q) does not matter in this definition of limit.
For such a multivariable limit to exist, this definition requires the value off approachesL along every possible path approaching(p,q).[17] In the above example, the functionsatisfies this condition. This can be seen by considering thepolar coordinateswhich givesHereθ =θ(r) is a function ofr which controls the shape of the path along whichf is approaching(p,q). Sincecosθ is bounded between [−1, 1], by thesandwich theorem, this limit tends to 0.
In contrast, the functiondoes not have a limit at(0, 0). Taking the path(x,y) = (t, 0) → (0, 0), we obtainwhile taking the path(x,y) = (t,t) → (0, 0), we obtain
Since the two values do not agree,f does not tend to a single value as(x,y) approaches(0, 0).
Although less commonly used, there is another type of limit for a multivariable function, known as themultiple limit. For a two-variable function, this is thedouble limit.[18] Let be defined on we saythe double limit off asx approachesp andy approachesq isL, written
if the following condition holds:
For such a double limit to exist, this definition requires the value off approachesL along every possible path approaching(p,q), excluding the two linesx =p andy =q. As a result, the multiple limit is a weaker notion than the ordinary limit: if the ordinary limit exists and equalsL, then the multiple limit exists and also equalsL. The converse is not true: the existence of the multiple limits does not imply the existence of the ordinary limit. Consider the examplewherebut does not exist.
If the domain off is restricted to then the two definitions of limits coincide.[18]
The concept of multiple limit can extend to the limit at infinity, in a way similar to that of a single variable function. For we saythe double limit off asx andy approaches infinity isL, written
if the following condition holds:
We saythe double limit off asx andy approaches minus infinity isL, written
if the following condition holds:
Let Instead of taking limit as(x,y) → (p,q), we may consider taking the limit of just one variable, say,x →p, to obtain a single-variable function ofy, namely In fact, this limiting process can be done in two distinct ways. The first one is calledpointwise limit. We saythe pointwise limit off asx approachesp isg, denoted or
Alternatively, we may sayf tends tog pointwise asx approachesp, denoted or
This limit exists if the following holds:
Here,δ =δ(ε,y) is a function of bothε andy. Eachδ is chosen for aspecific point ofy. Hence we say the limit is pointwise iny. For example,has a pointwise limit of constant zero functionbecause for every fixedy, the limit is clearly 0. This argument fails ify is not fixed: ify is very close toπ/2, the value of the fraction may deviate from 0.
This leads to another definition of limit, namely theuniform limit. We saythe uniform limit off onT asx approachesp isg, denoted or
Alternatively, we may sayf tends tog uniformly onT asx approachesp, denoted or
This limit exists if the following holds:
Here,δ =δ(ε) is a function of onlyε but noty. In other words,δ isuniformly applicable to ally inT. Hence we say the limit is uniform iny. For example,has a uniform limit of constant zero functionbecause for all realy,cosy is bounded between[−1, 1]. Hence no matter howy behaves, we may use thesandwich theorem to show that the limit is 0.
Let We may consider taking the limit of just one variable, say,x →p, to obtain a single-variable function ofy, namely and then take limit in the other variable, namelyy →q, to get a numberL. Symbolically,
This limit is known asiterated limit of the multivariable function.[20] The order of taking limits may affect the result, i.e.,
in general.
A sufficient condition of equality is given by theMoore-Osgood theorem, which requires the limit to be uniform onT.[21]
SupposeM andN are subsets ofmetric spacesA andB, respectively, andf :M →N is defined betweenM andN, withx ∈M,p alimit point ofM andL ∈N. It is said thatthe limit off asx approachesp isL and write
if the following property holds:
Again, note thatp need not be in the domain off, nor doesL need to be in the range off, and even iff(p) is defined it need not be equal toL.
The limit inEuclidean space is a direct generalization of limits tovector-valued functions. For example, we may consider a function such thatThen, under the usualEuclidean metric,if the following holds:
In this example, the function concerned are finite-dimension vector-valued function. In this case, thelimit theorem for vector-valued function states that if the limit of each component exists, then the limit of a vector-valued function equals the vector with each component taken the limit:[23]
One might also want to consider spaces other than Euclidean space. An example would be the Manhattan space. Consider such thatThen, under theManhattan metric,if the following holds:
Since this is also a finite-dimension vector-valued function, the limit theorem stated above also applies.[24]
Finally, we will discuss the limit infunction space, which has infinite dimensions. Consider a functionf(x,y) in the function space We want to find out asx approachesp, howf(x,y) will tend to another functiong(y), which is in the function space The "closeness" in this function space may be measured under theuniform metric.[25] Then, we will saythe uniform limit off onT asx approachesp isg and write or
if the following holds:
In fact, one can see that this definition is equivalent to that of the uniform limit of a multivariable function introduced in the previous section.
Suppose and aretopological spaces with aHausdorff space. Let be alimit point of, and. For a function, it is said that thelimit of as approaches is, written
if the following property holds:
This last part of the definition can also be phrased as "there exists an openpunctured neighbourhood of such that.
The domain of does not need to contain. If it does, then the value of at is irrelevant to the definition of the limit. In particular, if the domain of is (or all of), then the limit of as exists and is equal toL if, for all subsetsΩ ofX with limit point, the limit of the restriction of toΩ exists and is equal toL. Sometimes this criterion is used to establish thenon-existence of the two-sided limit of a function on by showing that theone-sided limits either fail to exist or do not agree. Such a view is fundamental in the field ofgeneral topology, where limits and continuity at a point are defined in terms of special families of subsets, calledfilters, or generalized sequences known asnets.
Alternatively, the requirement that be a Hausdorff space can be relaxed to the assumption that be a general topological space, but then the limit of a function may not be unique. In particular, one can no longer talk aboutthe limit of a function at a point, but rathera limit orthe set of limits at a point.
A function is continuous at a limit point of and in its domain if and only if isthe (or, in the general case,a) limit of as tends to.
There is another type of limit of a function, namely thesequential limit. Let be a mapping from a topological spaceX into a Hausdorff spaceY, a limit point ofX andL ∈Y. The sequential limit of as tends to isL if
IfL is the limit (in the sense above) of as approaches, then it is a sequential limit as well; however, the converse need not hold in general. If in additionX ismetrizable, thenL is the sequential limit of as approaches if and only if it is the limit (in the sense above) of as approaches.
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.) In this setting:if, and only if, for all sequencesxn (with, for alln,xn not equal toa) converging toa the sequencef(xn) converges toL. It was shown bySierpiński in 1916 that proving the equivalence of this definition and the definition above, requires and is equivalent to a weak form of theaxiom of choice. Note that defining what it means for a sequencexn to converge toa requires theepsilon, delta method.
Similarly as it was the case of Weierstrass's definition, a more general Heine definition applies to functions defined onsubsets of the real line. Letf be a real-valued function with the domainDm(f ). Leta be the limit of a sequence of elements ofDm(f ) \ {a}. Then the limit (in this sense) off isL asx approachesa if for every sequencexn ∈Dm(f ) \ {a} (so that for alln,xn is not equal toa) that converges toa, the sequencef(xn) converges toL. This is the same as the definition of a sequential limit in the preceding section obtained by regarding the subsetDm(f ) of as a metric space with the induced metric.
In non-standard calculus the limit of a function is defined by:if and only if for all is infinitesimal wheneverx −a is infinitesimal. Here are thehyperreal numbers andf* is the natural extension off to the non-standard real numbers.Keisler proved that such a hyperrealdefinition of limit reduces the quantifier complexity by two quantifiers.[26] 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.[27] Bŀaszczyk et al. detail the usefulness ofmicrocontinuity in developing a transparent definition ofuniform continuity, and characterize Hrbacek's criticism as a "dubious lament".[28]
At the 1908 international congress of mathematicsF. Riesz introduced an alternate way defining limits and continuity in concept called "nearness".[29] A pointx is defined to be near a set if for everyr > 0 there is a pointa ∈A so that|x −a| <r. In this setting theif and only if for allL is nearf(A) whenevera is nearA.Heref(A) is the set This definition can also be extended to metric and topological spaces.
The notion of the limit of a function is very closely related to the concept of continuity. A functionf is said to becontinuous atc if it is both defined atc and its value atc equals the limit off asx approachesc:
We have here assumed thatc is alimit point of the domain off.
If a functionf is real-valued, then the limit off atp isL if and only if both the right-handed limit and left-handed limit off atp exist and are equal toL.[30]
The functionf iscontinuous atp if and only if the limit off(x) asx approachesp exists and is equal tof(p). Iff :M →N is a function between metric spacesM andN, then it is equivalent thatf transforms every sequence inM which converges towardsp into a sequence inN which converges towardsf(p).
IfN is anormed vector space, then the limit operation is linear in the following sense: if the limit off(x) asx approachesp isL and the limit ofg(x) asx approachesp isP, then the limit off(x) + g(x) asx approachesp isL +P. Ifa is a scalar from the basefield, then the limit ofaf(x) asx approachesp isaL.
Iff andg are real-valued (or complex-valued) functions, then taking the limit of an operation onf(x) andg(x) (e.g.,f +g,f −g,f ×g,f /g,f g) under certain conditions is compatible with the operation of limits off(x) andg(x). This fact is often called thealgebraic 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).
These rules are also valid for one-sided limits, including whenp 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.
(see alsoExtended real number line).
In other cases the limit on the left may still exist, although the right-hand side, called anindeterminate form, does not allow one to determine the result. This depends on the functionsf andg. These indeterminate forms are:
See furtherL'Hôpital's rule below andIndeterminate form.
In general, from knowing that and it doesnot follow that However, this "chain rule" does hold if one of the followingadditional conditions holds:
As an example of this phenomenon, consider the following function that violates both additional restrictions:
Since the value atf(0) is aremovable discontinuity, for alla.Thus, the naïve chain rule would suggest that the limit off(f(x)) is 0. However, it is the case thatand so for alla.
Forn a nonnegative integer and constants and
This can be proven by dividing both the numerator and denominator byxn. 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.
This rule usesderivatives to find limits ofindeterminate forms0/0 or±∞/∞, and only applies to such cases. Other indeterminate forms may be manipulated into this form. Given two functionsf(x) andg(x), defined over anopen intervalI containing the desired limit pointc, then if:
then:
Normally, the first condition is the most important one.
For example:
Specifying an infinite bound on a summation or integral is a common shorthand for specifying a limit.
A short way to write the limitis An important example of limits of sums such as these areseries.
A short way to write the limitis
A short way to write the limitis
