{"history":[],"article":{"id":"097c6fa1-c462-40b4-8484-57115359ee63","title":"Limit of a function","slug":"limit-of-a-function","content":":::summary\n:::\n\n# \nAlthough the function \\tfrac{\\sin x}{x} is not defined at zero, as x becomes closer and closer to zero, \\tfrac{\\sin x}{x} becomes arbitrarily close to 1.\nIn other words, the limit of \\tfrac{\\sin x}{x}, as x approaches zero, equals 1.\nIn [mathematics](https://bliptext.com/articles/mathematics), the **limit of a [function](https://bliptext.com/articles/function-mathematics)** is a fundamental concept in [calculus](https://bliptext.com/articles/calculus) and [analysis](https://bliptext.com/articles/mathematical-analysis) concerning the behavior of that [function](https://bliptext.com/articles/function-mathematics) near a particular [input](https://bliptext.com/articles/independent-variable) which may or may not be in the [domain](https://bliptext.com/articles/domain-mathematical-analysis) of the [function](https://bliptext.com/articles/function-mathematics).\nFormal definitions, first devised in the early 19th century, are given below.\nInformally, a function f assigns an [output](https://bliptext.com/articles/dependent-variable)\nf(x)\nto every input x. We say that the function has a limit L at an input p, if\nf(x)\ngets closer and closer to L as x moves closer and closer to p. More specifically, the output value can be made arbitrarily close to L if the input to f is taken sufficiently close to p. 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.\nThe notion of a limit has many applications in [modern calculus](https://bliptext.com/articles/calculus).\nIn particular, the many definitions of [continuity](https://bliptext.com/articles/continuous-function) employ the concept of limit: roughly, a function is continuous if all of its limits agree with the values of the function.\nThe concept of limit also appears in the definition of the [derivative](https://bliptext.com/articles/derivative): in the calculus of one variable, this is the limiting value of the [slope](https://bliptext.com/articles/slope) of [secant lines](https://bliptext.com/articles/secant-line) to the graph of a function.\n\n# History\nAlthough implicit in the [development of calculus](https://bliptext.com/articles/history-of-calculus) of the 17th and 18th centuries, the modern idea of the limit of a function goes back to [Bolzano](https://bliptext.com/articles/bernard-bolzano) who, in 1817, introduced the basics of the epsilon-delta technique (see [(ε, δ)-definition of limit](https://bliptext.com/articles/) below) to define continuous functions.\nHowever, his work was not known during his lifetime.\nIn his 1821 book Cours d'analyse, [Augustin-Louis Cauchy](https://bliptext.com/articles/augustin-louis-cauchy) discussed variable quantities, [infinitesimals](https://bliptext.com/articles/infinitesimal) and limits, and defined continuity of y=f(x) by saying that an infinitesimal change in x necessarily produces an infinitesimal change in y, while Grabiner claims that he used a rigorous epsilon-delta definition in proofs.\nIn 1861, [Weierstrass](https://bliptext.com/articles/weierstrass) first introduced the epsilon-delta definition of limit in the form it is usually written today.\nHe also introduced the notations \\lim and\nThe modern notation of placing the arrow below the limit symbol is due to [Hardy](https://bliptext.com/articles/g-h-hardy), which is introduced in his book [A Course of Pure Mathematics](https://bliptext.com/articles/a-course-of-pure-mathematics) in 1908.\n\n# Motivation\nImagine a person walking on a landscape represented by the graph\ny = f(x)\n. Their horizontal position is given by x, much like the position given by a map of the land or by a [global positioning system](https://bliptext.com/articles/global-positioning-system).\nTheir altitude is given by the coordinate y. Suppose they walk towards a position\nx = p\n, as they get closer and closer to this point, they will notice that their altitude approaches a specific value L. If asked about the altitude corresponding to\nx = p\n, they would reply by saying\ny = L\n.\nWhat, 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.\nFor 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, arguing that as long as they are within fifty horizontal meters of p, their altitude is always within ten meters of L.\nThe accuracy goal is then changed: can they get within one vertical meter?\nYes, supposing that they are able to move within five horizontal meters of p, their altitude will always remain within one meter from the target altitude L. Summarizing the aforementioned concept we can say that the traveler's altitude approaches L as their horizontal position approaches p, so as to say that for every target accuracy goal, however small it may be, there is some neighbourhood of p where all (not just some) altitudes corresponding to all the horizontal positions in that neighbourhood, except maybe for the horizontal position p itself, fulfill that accuracy goal.\nThe initial informal statement can now be explicated:\nIn fact, this explicit statement is quite close to the formal definition of the limit of a function, with values in a [topological space](https://bliptext.com/articles/hausdorff-space).\nMore specifically, to say that\nis to say that\nf(x)\ncan be made as close to L as desired, by making x close enough, but not equal, to p.\nThe following definitions, known as\n(ε, δ)\n-definitions, are the generally accepted definitions for the limit of a function in various contexts.\n\n# Functions of a single variable\n\n## (ε, δ)\n\n-definition of limit\n[[File:Epsilon-delta limit.svg|thumb|For the depicted f, a, and b, we can ensure that the value\nf(x)\nis within an arbitrarily small interval\n(b – ε, b + ε)\nby restricting x to a sufficiently small interval\n(a – δ, a + δ).\nHence\nf(x) → b\nas\nx → a\n.]] Suppose is a function defined on the [real line](https://bliptext.com/articles/real-line), and there are two real numbers p and L. One would say that **the limit of f, as x approaches p, is L** and written\nor alternatively, say '''\nf(x)\ntends to L as x tends to p''', and written:\nif the following property holds: for every real\nε > 0\n, there exists a real\nδ > 0\nsuch that for all real x,\n0 <\nimplies\n. Symbolically,\nFor example, we may say\nbecause for every real\nε > 0\n, we can take\nδ = ε/4\n, so that for all real x, if\n0 <\n, then\n.\nA more general definition applies for functions defined on [subsets](https://bliptext.com/articles/subset) of the real line.\nLet S be a subset of \\R.\nLet f: S \\to \\R be a [real-valued function](https://bliptext.com/articles/real-valued-function).\nLet p be a point such that there exists some open interval\n(a, b)\ncontaining p with It is then said that the limit of f as x approaches p is L, if:\nOr, symbolically:\nFor example, we may say\nbecause for every real\nε > 0\n, we can take\nδ = ε\n, so that for all real\nx ≥ −3\n, if\n0 <\n, then\n. In this example,\nS = [−3, ∞)\ncontains open intervals around the point 1 (for example, the interval (0, 2)).\nHere, note that the value of the limit does not depend on f being defined at p, nor on the value\nf(p)\n—if it is defined.\nFor example, let\nbecause for every\nε > 0\n, we can take\nδ = ε/2\n, so that for all real\nx ≠ 1\n, if\n0 <\n, then\n. Note that here\nf(1)\nis undefined.\nIn fact, a limit can exist in which equals where\nint S\nis the [interior](https://bliptext.com/articles/interior-topology) of S, and\niso Sc\nare the [isolated points](https://bliptext.com/articles/isolated-point) of the complement of S. In our previous example where We see, specifically, this definition of limit allows a limit to exist at 1, but not 0 or 2.\nThe letters ε and δ can be understood as \"error\" and \"distance\".\nIn 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](https://bliptext.com/articles/cours-d-analyse)).\nIn 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.\nAs discussed below, this definition also works for functions in a more general context.\nThe idea that δ and ε represent distances helps suggest these generalizations.\n\n## Existence and one-sided limits\n[[Image:Upper semi.svg|thumb|The limit as x \\to x_0^+ differs from that as Therefore, the limit as\nx → x0\ndoes not exist.]]\nAlternatively, x may approach p from above (right) or below (left), in which case the limits may be written as\nor\nrespectively.\nIf these limits exist at p and are equal there, then this can be referred to as the limit of\nf(x)\nat 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).\nIf either one-sided limit does not exist at p, then the limit at p also does not exist.\nA formal definition is as follows.\nThe **limit of f as x approaches p from above is L** if:\nε > 0\n, there exists a\nδ > 0\nsuch that whenever\n0 < x − p < δ\n, we have\n.\nThe **limit of f as x approaches p from below is L** if:\nε > 0\n, there exists a\nδ > 0\nsuch that whenever\n0 < p − x < δ\n, we have\n.\nIf the limit does not exist, then the [oscillation](https://bliptext.com/articles/oscillation-of-a-function-at-a-point) of f at p is non-zero.\n# [The first three functions have points for which the limit does not exist, while the functionis not defined at x = 0, but its limit does exist. | upload.wikimedia.org/wikipedia/commons/5/58/Undefined///limit///examples.png]\n\n## More general definition using limit points and subsets\nLimits can also be defined by approaching from subsets of the domain.\nIn general: Let be a real-valued function defined on some Let p be a [limit point](https://bliptext.com/articles/limit-point) of some T \\subset S—that is, p is the limit of some sequence of elements of T distinct from p. Then we say **the limit of f, as x approaches p from values in T, is L**, written\nif the following holds:\nNote, T can be any subset of S, the domain of f. And the limit might depend on the selection of T. This generalization includes as special cases limits on an interval, as well as left-handed limits of real-valued functions (e.g., by taking T to be an open interval of the form\n(–∞, a)\n), and right-handed limits (e.g., by taking T to be an open interval of the form\n(a, ∞)\n). It also extends the notion of one-sided limits to the included endpoints of (half-)closed intervals, so the [square root function](https://bliptext.com/articles/square-root-function) can have limit 0 as x approaches 0 from above:\nsince for every\nε > 0\n, we may take\nδ = ε\nsuch that for all\nx ≥ 0\n, if\n0 <\n, then\n.\nThis definition allows a limit to be defined at limit points of the domain S, if a suitable subset T which has the same limit point is chosen.\nNotably, the previous two-sided definition works on which is a subset of the limit points of S.\nFor example, let The previous two-sided definition would work at but it wouldn't work at 0 or 2, which are limit points of S.\n\n## Deleted versus non-deleted limits\nThe definition of limit given here does not depend on how (or whether) f is defined at p. Bartle 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. Let be a real-valued function.\n**The non-deleted limit of f, as x approaches p, is L** if\nThe definition is the same, except that the neighborhood\nnow includes the point p, in contrast to the [deleted neighborhood](https://bliptext.com/articles/deleted-neighborhood)\n0 <\n. This makes the definition of a non-deleted limit less general.\nOne of the advantages of working with non-deleted limits is that they allow to state the [theorem about limits of compositions](https://bliptext.com/articles/) without any constraints on the functions (other than the existence of their non-deleted limits).\nBartle notes that although by \"limit\" some authors do mean this non-deleted limit, deleted limits are the most popular.\n\n## Examples\n\n### Non-existence of one-sided limit(s)\nThe function\nhas no limit at\nx0 = 1\n(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 other x-coordinate.\nThe function\n(a.k.a., the [Dirichlet function](https://bliptext.com/articles/dirichlet-function)) has no limit at any x-coordinate.\n# [Function without a limit at an essential discontinuity | upload.wikimedia.org/wikipedia/commons/9/92/Discontinuity///essential.svg]\n\n### Non-equality of one-sided limits\nThe function\nhas a limit at every non-zero x-coordinate (the limit equals 1 for negative x and equals 2 for positive x).\nThe limit at\nx = 0\ndoes not exist (the left-hand limit equals 1, whereas the right-hand limit equals 2).\n\n### Limits at only one point\nThe functions\nand\nboth have a limit at\nx = 0\nand it equals 0.\n\n### Limits at countably many points\nThe function\nhas a limit at any x-coordinate of the form where n is any integer.\n\n# Limits involving infinity\n\n## Limits at infinity\nLet f:S \\to \\R be a function defined on **The limit of f as x approaches infinity is L**, denoted\nmeans that:\nSimilarly, **the limit of f as x approaches minus infinity is L**, denoted\nmeans that:\nFor example,\nbecause for every\nε > 0\n, we can take\nc = 3/ε\nsuch that for all real x, if\nx > c\n, then\n.\nAnother example is that\nbecause for every\nε > 0\n, we can take\nc = max{1, −ln(ε)}\nsuch that for all real x, if\nx < −c\n, then\n.\n# [The limit of this function at infinity exists | upload.wikimedia.org/wikipedia/commons/b/b2/Limit///Infinity///SVG.svg]\n\n## Infinite limits\nFor a function whose values grow without bound, the function diverges and the usual limit does not exist.\nHowever, in this case one may introduce limits with infinite values.\nLet be a function defined on The statement **the limit of f as x approaches p is infinity**, denoted\nmeans that:\nThe statement **the limit of f as x approaches p is minus infinity**, denoted\nmeans that:\nFor example,\nbecause for every\nN > 0\n, we can take such that for all real\nx > 0\n, if\n0 < x − 1 < δ\n, then\nf(x) > N\n.\nThese ideas can be used together to produce definitions for different combinations, such as\nor\nFor example,\nbecause for every\nN > 0\n, we can take\nδ = e−N\nsuch that for all real\nx > 0\n, if\n0 < x − 0 < δ\n, then\nf(x) < −N\n.\nLimits involving infinity are connected with the concept of [asymptotes](https://bliptext.com/articles/asymptote).\nThese notions of a limit attempt to provide a metric space interpretation to limits at infinity.\nIn fact, they are consistent with the topological space definition of limit if\n[−∞, c)\nfor some c \\in \\R,\n(c, ∞]\nwhere c \\in \\R, and\nIn this case, \\overline \\R is a topological space and any function of the form f : X \\to Y with is subject to the topological definition of a limit.\nNote 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.\n\n## Alternative notation\nMany authors allow for the [projectively extended real line](https://bliptext.com/articles/projectively-extended-real-line) to be used as a way to include infinite values as well as [extended real line](https://bliptext.com/articles/extended-real-number-line).\nWith this notation, the extended real line is given as \\R \\cup \\{-\\infty, +\\infty\\} and the projectively extended real line is \\R \\cup \\{\\infty\\} 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.\nAs 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 +∞).\nThere are also noteworthy pitfalls.\nFor example, when working with the extended real line, x^{-1} does not possess a central limit (which is normal):\nIn contrast, when working with the projective real line, infinities (much like 0) are unsigned, so, the central limit does exist in that context:\nIn fact there are a plethora of conflicting formal systems in use.\nIn certain applications of [numerical differentiation and integration](https://bliptext.com/articles/numerical-analysis), it is, for example, convenient to have [signed zeroes](https://bliptext.com/articles/negative-zero).\nA simple reason has to do with the converse of namely, it is convenient for to be considered true.\nSuch zeroes can be seen as an approximation to [infinitesimals](https://bliptext.com/articles/infinitesimal).\n\n## Limits at infinity for rational functions\n[[File:Tamasol SVG.svg|thumb|300px|Horizontal asymptote about\ny = 4\n]] There are three basic rules for evaluating limits at infinity for a [rational function](https://bliptext.com/articles/rational-function) (where p and q are polynomials):\nIf the limit at infinity exists, it represents a horizontal asymptote at\ny = L\n. Polynomials do not have horizontal asymptotes; such asymptotes may however occur with rational functions.\n\n# Functions of more than one variable\n\n## Ordinary limits\nBy noting that\nrepresents a [distance](https://bliptext.com/articles/distance), the definition of a limit can be extended to functions of more than one variable.\nIn the case of a function defined on we defined the limit as follows: '''the limit of f as\n(x, y)\napproaches\n(p, q)\nis L''', written\nif the following condition holds:\nε > 0\n, there exists a\nδ > 0\nsuch that for all x in S and y in T, whenever we have\n,\nor formally:\nHere is the [Euclidean distance](https://bliptext.com/articles/euclidean-distance) between\n(x, y)\nand\n(p, q)\n. (This can in fact be replaced by any [norm](https://bliptext.com/articles/norm-mathematics)\n, and be extended to any number of variables.)\nFor example, we may say\nbecause for every\nε > 0\n, we can take such that for all real\nx ≠ 0\nand real\ny ≠ 0\n, if then\n.\nSimilar to the case in single variable, the value of f at\n(p, q)\ndoes not matter in this definition of limit.\nFor such a multivariable limit to exist, this definition requires the value of f approaches L along every possible path approaching\n(p, q)\n. In the above example, the function\nsatisfies this condition.\nThis can be seen by considering the [polar coordinates](https://bliptext.com/articles/polar-coordinates)\nwhich gives\nHere\nθ = θ(r)\nis a function of r which controls the shape of the path along which f is approaching\n(p, q)\n. Since\ncos θ\nis bounded between [−1, 1], by the [sandwich theorem](https://bliptext.com/articles/sandwich-theorem), this limit tends to 0.\nIn contrast, the function\ndoes not have a limit at\n(0, 0)\n. Taking the path\n(x, y) = (t, 0) → (0, 0)\n, we obtain\nwhile taking the path\n(x, y) = (t, t) → (0, 0)\n, we obtain\nSince the two values do not agree, f does not tend to a single value as\n(x, y)\napproaches\n(0, 0)\n.\n\n## Multiple limits\nAlthough less commonly used, there is another type of limit for a multivariable function, known as the **multiple limit**.\nFor a two-variable function, this is the **double limit**.\nLet be defined on we say **the double limit of f as x approaches p and y approaches q is L**, written\nif the following condition holds:\nFor such a double limit to exist, this definition requires the value of f approaches L along every possible path approaching\n(p, q)\n, excluding the two lines\nx = p\nand\ny = q\n. As a result, the multiple limit is a weaker notion than the ordinary limit: if the ordinary limit exists and equals L, then the multiple limit exists and also equals L. The converse is not true: the existence of the multiple limits does not imply the existence of the ordinary limit.\nConsider the example\nwhere\nbut\ndoes not exist.\nIf the domain of f is restricted to then the two definitions of limits coincide.\n\n## Multiple limits at infinity\nThe concept of multiple limit can extend to the limit at infinity, in a way similar to that of a single variable function.\nFor we say **the double limit of f as x and y approaches infinity is L**, written\nif the following condition holds:\nWe say **the double limit of f as x and y approaches minus infinity is L**, written\nif the following condition holds:\n\n## Pointwise limits and uniform limits\nLet Instead of taking limit as\n(x, y) → (p, q)\n, we may consider taking the limit of just one variable, say,\nx → p\n, to obtain a single-variable function of y, namely In fact, this limiting process can be done in two distinct ways.\nThe first one is called **pointwise limit**.\nWe say **the pointwise limit of f as x approaches p is g**, denoted\nor\nAlternatively, we may say **f tends to g pointwise as x approaches p**, denoted\nor\nThis limit exists if the following holds:\nHere,\nδ = δ(ε, y)\nis a function of both ε and y. Each δ is chosen for a specific point of y. Hence we say the limit is pointwise in y. For example,\nhas a pointwise limit of constant zero function\nbecause for every fixed y, the limit is clearly 0.\nThis argument fails if y is not fixed: if y is very close to\nπ/2\n, the value of the fraction may deviate from 0.\nThis leads to another definition of limit, namely the **uniform limit**.\nWe say **the uniform limit of f on T as x approaches p is g**, denoted\nor\nAlternatively, we may say **f tends to g uniformly on T as x approaches p**, denoted\nor\nThis limit exists if the following holds:\nHere,\nδ = δ(ε)\nis a function of only ε but not y. In other words, δ is uniformly applicable to all y in T. Hence we say the limit is uniform in y. For example,\nhas a uniform limit of constant zero function\nbecause for all real y,\ncos y\nis bounded between\n[−1, 1]\n. Hence no matter how y behaves, we may use the [sandwich theorem](https://bliptext.com/articles/sandwich-theorem) to show that the limit is 0.\n\n## Iterated limits\nLet We may consider taking the limit of just one variable, say,\nx → p\n, to obtain a single-variable function of y, namely and then take limit in the other variable, namely\ny → q\n, to get a number L. Symbolically,\nThis limit is known as **iterated limit** of the multivariable function.\nThe order of taking limits may affect the result, i.e.,\nin general.\nA sufficient condition of equality is given by the [Moore-Osgood theorem](https://bliptext.com/articles/moore-osgood-theorem), which requires the limit to be uniform on T.\n\n# Functions on metric spaces\nSuppose M and N are subsets of [metric spaces](https://bliptext.com/articles/metric-spaces) A and B, respectively, and\nf : M → N\nis defined between M and N, with\nx ∈ M\n, p a [limit point](https://bliptext.com/articles/limit-point) of M and\nL ∈ N\n. It is said that **the limit of f as x approaches p is L** and write\nif the following property holds:\nAgain, 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\nf(p)\nis defined it need not be equal to L.\n\n## Euclidean metric\nThe limit in [Euclidean space](https://bliptext.com/articles/euclidean-space) is a direct generalization of limits to [vector-valued functions](https://bliptext.com/articles/vector-valued-functions).\nFor example, we may consider a function such that\nThen, under the usual [Euclidean metric](https://bliptext.com/articles/euclidean-metric),\nif the following holds:\nIn this example, the function concerned are finite-[dimension](https://bliptext.com/articles/dimension-vector-space) vector-valued function.\nIn this case, the **limit 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:\n\n## Manhattan metric\nOne might also want to consider spaces other than Euclidean space.\nAn example would be the Manhattan space.\nConsider such that\nThen, under the [Manhattan metric](https://bliptext.com/articles/manhattan-metric),\nif the following holds:\nSince this is also a finite-dimension vector-valued function, the limit theorem stated above also applies.\n\n## Uniform metric\nFinally, we will discuss the limit in [function space](https://bliptext.com/articles/function-space), which has infinite dimensions.\nConsider a function\nf(x, y)\nin the function space We want to find out as x approaches p, how\nf(x, y)\nwill tend to another function\ng(y)\n, which is in the function space T \\to \\R.\nThe \"closeness\" in this function space may be measured under the [uniform metric](https://bliptext.com/articles/uniform-metric).\nThen, we will say **the uniform limit of f on T as x approaches p is g** and write\nor\nif the following holds:\nIn fact, one can see that this definition is equivalent to that of the uniform limit of a multivariable function introduced in the previous section.\n\n# Functions on topological spaces\nSuppose X and Y are [topological spaces](https://bliptext.com/articles/topological-space) with Y a [Hausdorff space](https://bliptext.com/articles/hausdorff-space).\nLet p be a [limit point](https://bliptext.com/articles/limit-point) of\nΩ ⊆ X\n, and\nL ∈ Y\n. For a function\nf : Ω → Y\n, it is said that the **limit of f as x approaches p is L**, written\nif the following property holds:\nThis last part of the definition can also be phrased \"there exists an open [punctured neighbourhood](https://bliptext.com/articles/neighbourhood-mathematics) U of p such that\nf(U ∩ Ω) ⊆ V\n\".\nThe 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.\nIn particular, if the domain of f is\nX − {p}\n(or all of X), then the limit of f as\nx → p\nexists and is equal to L if, for all subsets\nΩ\nof X with limit point p, the limit of the restriction of f to\nΩ\nexists 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 limits](https://bliptext.com/articles/one-sided-limit) either fail to exist or do not agree.\nSuch a view is fundamental in the field of [general topology](https://bliptext.com/articles/general-topology), where limits and continuity at a point are defined in terms of special families of subsets, called [filters](https://bliptext.com/articles/filter-set-theory), or generalized sequences known as [nets](https://bliptext.com/articles/net-mathematics).\nAlternatively, 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.\nIn 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.\nA function is continuous at a limit point p of and in its domain if and only if\nf(p)\nis the (or, in the general case, a) limit of\nf(x)\nas x tends to p.\nThere is another type of limit of a function, namely the **sequential limit**.\nLet\nf : X → Y\nbe a mapping from a topological space X into a Hausdorff space Y,\np ∈ X\na limit point of X and\nL ∈ Y\n. The sequential limit of f as x tends to p is L if\nX − {p}\nthat [converges](https://bliptext.com/articles/limit-of-a-sequence) to p, the sequence\nf(xn)\n[converges](https://bliptext.com/articles/limit-of-a-sequence) to L.\nIf 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.\nIf in addition X is [metrizable](https://bliptext.com/articles/metrizable), 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.\n\n# Other characterizations\n\n## In terms of sequences\nFor functions on the real line, one way to define the limit of a function is in terms of the limit of sequences.\n(This definition is usually attributed to [Eduard Heine](https://bliptext.com/articles/eduard-heine).) In this setting:\nif, and only if, for all sequences xn (with xn not equal to a for all n) converging to a the sequence\nf(xn)\nconverges to L. It was shown by [Sierpiński](https://bliptext.com/articles/sierpi-ski) 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](https://bliptext.com/articles/axiom-of-choice).\nNote that defining what it means for a sequence xn to converge to a requires the [epsilon, delta method](https://bliptext.com/articles/epsilon-delta-method).\nSimilarly as it was the case of Weierstrass's definition, a more general Heine definition applies to functions defined on [subsets](https://bliptext.com/articles/subset) of the real line.\nLet f be a real-valued function with the domain\nDm(f )\n. Let a be the limit of a sequence of elements of\nDm(f ) \\ {a}.\nThen the limit (in this sense) of f is L as x approaches p if for every sequence\nxn ∈ Dm(f ) \\ {a}\n(so that for all n, xn is not equal to a) that converges to a, the sequence\nf(xn)\nconverges to L. This is the same as the definition of a sequential limit in the preceding section obtained by regarding the subset\nDm(f )\nof \\R as a metric space with the induced metric.\n\n## In non-standard calculus\nIn non-standard calculus the limit of a function is defined by:\nif and only if for all x\\in \\R^*, f^*(x)-L is infinitesimal whenever\nx − a\nis infinitesimal.\nHere \\R^* are the [hyperreal numbers](https://bliptext.com/articles/hyperreal-number) and f* is the natural extension of f to the non-standard real numbers.\n[Keisler](https://bliptext.com/articles/howard-jerome-keisler) proved that such a hyperreal [definition of limit](https://bliptext.com/articles/non-standard-calculus) reduces the quantifier complexity by two quantifiers.\nOn 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.\nBŀaszczyk et al. detail the usefulness of [microcontinuity](https://bliptext.com/articles/microcontinuity) in developing a transparent definition of uniform continuity, and characterize Hrbacek's criticism as a \"dubious lament\".\n\n## In terms of nearness\nAt the 1908 international congress of mathematics [F. Riesz](https://bliptext.com/articles/frigyes-riesz) introduced an alternate way defining limits and continuity in concept called \"nearness\".\nA point x is defined to be near a set if for every\nr > 0\nthere is a point\na ∈ A\nso that\n. In this setting the\nif and only if for all L is near\nf(A)\nwhenever a is near A. Here\nf(A)\nis the set This definition can also be extended to metric and topological spaces.\n\n# Relationship to continuity\nThe notion of the limit of a function is very closely related to the concept of continuity.\nA function f is said to be [continuous](https://bliptext.com/articles/continuous-function) at c if it is both defined at c and its value at c equals the limit of f as x approaches c:\nWe have here assumed that c is a [limit point](https://bliptext.com/articles/limit-point) of the domain of f.\n\n# Properties\nIf 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.\nThe function f is [continuous](https://bliptext.com/articles/continuous-function) at p if and only if the limit of\nf(x)\nas x approaches p exists and is equal to\nf(p)\n. If\nf : M → N\nis 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\nf(p)\n.\nIf N is a [normed vector space](https://bliptext.com/articles/normed-vector-space), then the limit operation is linear in the following sense: if the limit of\nf(x)\nas x approaches p is L and the limit of\ng(x)\nas x approaches p is P, then the limit of\nf(x) + g(x)\nas x approaches p is\nL + P\n. If a is a scalar from the base [field](https://bliptext.com/articles/field-mathematics), then the limit of\naf(x)\nas x approaches p is aL.\nIf f and g are real-valued (or complex-valued) functions, then taking the limit of an operation on\nf(x)\nand\ng(x)\n(e.g.,\nf + g\n,\nf − g\n,\nf × g\n,\nf / g\n, fg) under certain conditions is compatible with the operation of limits of\nf(x)\nand\ng(x)\n. This fact is often called the **algebraic limit theorem**.\nThe 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).\nAdditionally, 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).\nThese rules are also valid for one-sided limits, including when p is ∞ or −∞.\nIn 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.\n(see also [Extended real number line](https://bliptext.com/articles/extended-real-number-line)).\nIn other cases the limit on the left may still exist, although the right-hand side, called an [indeterminate form](https://bliptext.com/articles/indeterminate-form), does not allow one to determine the result.\nThis depends on the functions f and g. These indeterminate forms are:\nSee further [L'Hôpital's rule](https://bliptext.com/articles/) below and [Indeterminate form](https://bliptext.com/articles/indeterminate-form).\n\n## Limits of compositions of functions\nIn general, from knowing that and it does not follow that\nHowever, this \"chain rule\" does hold if one of the following additional conditions holds:\nf(b) = c\n(that is, f is continuous at b), or\nδ > 0\nsuch that if\n0 <\nthen\n).\nAs an example of this phenomenon, consider the following function that violates both additional restrictions:\nSince the value at\nf(0)\nis a [removable discontinuity](https://bliptext.com/articles/removable-discontinuity),\nfor all a. Thus, the naïve chain rule would suggest that the limit of\nf(f(x))\nis 0.\nHowever, it is the case that\nand so for all a.\n\n## Limits of special interest\n\n### Rational functions\nFor n a nonnegative integer and constants and\nThis can be proven by dividing both the numerator and denominator by xn.\nIf the numerator is a polynomial of higher degree, the limit does not exist.\nIf the denominator is of higher degree, the limit is 0.\n\n### Trigonometric functions\n\n### Exponential functions\n\n### Logarithmic functions\n\n## L'Hôpital's rule\nThis rule uses [derivatives](https://bliptext.com/articles/derivative) to find limits of [indeterminate forms](https://bliptext.com/articles/indeterminate-forms)\n0/0\nor\n±∞/∞\n, and only applies to such cases.\nOther indeterminate forms may be manipulated into this form.\nGiven two functions\nf(x)\nand\ng(x)\n, defined over an [open interval](https://bliptext.com/articles/open-interval) I containing the desired limit point c, then if:\nthen:\nNormally, the first condition is the most important one.\nFor example:\n\n## Summations and integrals\nSpecifying an infinite bound on a summation or integral is a common shorthand for specifying a limit.\nA short way to write the limit\nis An important example of limits of sums such as these are [series](https://bliptext.com/articles/series-mathematics).\nA short way to write the limit\nis\nA short way to write the limit\nis","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:02:05.015Z","updated_at":"2025-01-25T19:02:05.015Z","search_vector":"'function':4 'limit':1","search_content":" \n \n\n# \nAlthough the function \\tfrac{\\sin x}{x} is not defined at zero, as x becomes closer and closer to zero, \\tfrac{\\sin x}{x} becomes arbitrarily close to 1.\nIn other words, the limit of \\tfrac{\\sin x}{x}, as x approaches zero, equals 1.\nIn [mathematics](https://bliptext.com/articles/mathematics), the **limit of a [function](https://bliptext.com/articles/function-mathematics)** is a fundamental concept in [calculus](https://bliptext.com/articles/calculus) and [analysis](https://bliptext.com/articles/mathematical-analysis) concerning the behavior of that [function](https://bliptext.com/articles/function-mathematics) near a particular [input](https://bliptext.com/articles/independent-variable) which may or may not be in the [domain](https://bliptext.com/articles/domain-mathematical-analysis) of the [function](https://bliptext.com/articles/function-mathematics).\nFormal definitions, first devised in the early 19th century, are given below.\nInformally, a function f assigns an [output](https://bliptext.com/articles/dependent-variable)\nf(x)\nto every input x. We say that the function has a limit L at an input p, if\nf(x)\ngets closer and closer to L as x moves closer and closer to p. More specifically, the output value can be made arbitrarily close to L if the input to f is taken sufficiently close to p. 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.\nThe notion of a limit has many applications in [modern calculus](https://bliptext.com/articles/calculus).\nIn particular, the many definitions of [continuity](https://bliptext.com/articles/continuous-function) employ the concept of limit: roughly, a function is continuous if all of its limits agree with the values of the function.\nThe concept of limit also appears in the definition of the [derivative](https://bliptext.com/articles/derivative): in the calculus of one variable, this is the limiting value of the [slope](https://bliptext.com/articles/slope) of [secant lines](https://bliptext.com/articles/secant-line) to the graph of a function.\n\n# History\nAlthough implicit in the [development of calculus](https://bliptext.com/articles/history-of-calculus) of the 17th and 18th centuries, the modern idea of the limit of a function goes back to [Bolzano](https://bliptext.com/articles/bernard-bolzano) who, in 1817, introduced the basics of the epsilon-delta technique (see [(ε, δ)-definition of limit](https://bliptext.com/articles/) below) to define continuous functions.\nHowever, his work was not known during his lifetime.\nIn his 1821 book Cours d'analyse, [Augustin-Louis Cauchy](https://bliptext.com/articles/augustin-louis-cauchy) discussed variable quantities, [infinitesimals](https://bliptext.com/articles/infinitesimal) and limits, and defined continuity of y=f(x) by saying that an infinitesimal change in x necessarily produces an infinitesimal change in y, while Grabiner claims that he used a rigorous epsilon-delta definition in proofs.\nIn 1861, [Weierstrass](https://bliptext.com/articles/weierstrass) first introduced the epsilon-delta definition of limit in the form it is usually written today.\nHe also introduced the notations \\lim and\nThe modern notation of placing the arrow below the limit symbol is due to [Hardy](https://bliptext.com/articles/g-h-hardy), which is introduced in his book [A Course of Pure Mathematics](https://bliptext.com/articles/a-course-of-pure-mathematics) in 1908.\n\n# Motivation\nImagine a person walking on a landscape represented by the graph\ny = f(x)\n. Their horizontal position is given by x, much like the position given by a map of the land or by a [global positioning system](https://bliptext.com/articles/global-positioning-system).\nTheir altitude is given by the coordinate y. Suppose they walk towards a position\nx = p\n, as they get closer and closer to this point, they will notice that their altitude approaches a specific value L. If asked about the altitude corresponding to\nx = p\n, they would reply by saying\ny = L\n.\nWhat, 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.\nFor 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, arguing that as long as they are within fifty horizontal meters of p, their altitude is always within ten meters of L.\nThe accuracy goal is then changed: can they get within one vertical meter?\nYes, supposing that they are able to move within five horizontal meters of p, their altitude will always remain within one meter from the target altitude L. Summarizing the aforementioned concept we can say that the traveler's altitude approaches L as their horizontal position approaches p, so as to say that for every target accuracy goal, however small it may be, there is some neighbourhood of p where all (not just some) altitudes corresponding to all the horizontal positions in that neighbourhood, except maybe for the horizontal position p itself, fulfill that accuracy goal.\nThe initial informal statement can now be explicated:\nIn fact, this explicit statement is quite close to the formal definition of the limit of a function, with values in a [topological space](https://bliptext.com/articles/hausdorff-space).\nMore specifically, to say that\nis to say that\nf(x)\ncan be made as close to L as desired, by making x close enough, but not equal, to p.\nThe following definitions, known as\n(ε, δ)\n-definitions, are the generally accepted definitions for the limit of a function in various contexts.\n\n# Functions of a single variable\n\n## (ε, δ)\n\n-definition of limit\n[[File:Epsilon-delta limit.svg|thumb|For the depicted f, a, and b, we can ensure that the value\nf(x)\nis within an arbitrarily small interval\n(b – ε, b + ε)\nby restricting x to a sufficiently small interval\n(a – δ, a + δ).\nHence\nf(x) → b\nas\nx → a\n.]] Suppose is a function defined on the [real line](https://bliptext.com/articles/real-line), and there are two real numbers p and L. One would say that **the limit of f, as x approaches p, is L** and written\nor alternatively, say '''\nf(x)\ntends to L as x tends to p''', and written:\nif the following property holds: for every real\nε > 0\n, there exists a real\nδ > 0\nsuch that for all real x,\n0 0\n, we can take\nδ = ε/4\n, so that for all real x, if\n0 0\n, we can take\nδ = ε\n, so that for all real\nx ≥ −3\n, if\n0 0\n, we can take\nδ = ε/2\n, so that for all real\nx ≠ 1\n, if\n0 0\n, there exists a\nδ > 0\nsuch that whenever\n0 0\n, there exists a\nδ > 0\nsuch that whenever\n0 0\n, we may take\nδ = ε\nsuch that for all\nx ≥ 0\n, if\n0 0\n, we can take\nc = 3/ε\nsuch that for all real x, if\nx > c\n, then\n.\nAnother example is that\nbecause for every\nε > 0\n, we can take\nc = max{1, −ln(ε)}\nsuch that for all real x, if\nx 0\n, we can take such that for all real\nx > 0\n, if\n0 N\n.\nThese ideas can be used together to produce definitions for different combinations, such as\nor\nFor example,\nbecause for every\nN > 0\n, we can take\nδ = e−N\nsuch that for all real\nx > 0\n, if\n0 0\n, there exists a\nδ > 0\nsuch that for all x in S and y in T, whenever we have\n,\nor formally:\nHere is the [Euclidean distance](https://bliptext.com/articles/euclidean-distance) between\n(x, y)\nand\n(p, q)\n. (This can in fact be replaced by any [norm](https://bliptext.com/articles/norm-mathematics)\n, and be extended to any number of variables.)\nFor example, we may say\nbecause for every\nε > 0\n, we can take such that for all real\nx ≠ 0\nand real\ny ≠ 0\n, if then\n.\nSimilar to the case in single variable, the value of f at\n(p, q)\ndoes not matter in this definition of limit.\nFor such a multivariable limit to exist, this definition requires the value of f approaches L along every possible path approaching\n(p, q)\n. In the above example, the function\nsatisfies this condition.\nThis can be seen by considering the [polar coordinates](https://bliptext.com/articles/polar-coordinates)\nwhich gives\nHere\nθ = θ(r)\nis a function of r which controls the shape of the path along which f is approaching\n(p, q)\n. Since\ncos θ\nis bounded between [−1, 1], by the [sandwich theorem](https://bliptext.com/articles/sandwich-theorem), this limit tends to 0.\nIn contrast, the function\ndoes not have a limit at\n(0, 0)\n. Taking the path\n(x, y) = (t, 0) → (0, 0)\n, we obtain\nwhile taking the path\n(x, y) = (t, t) → (0, 0)\n, we obtain\nSince the two values do not agree, f does not tend to a single value as\n(x, y)\napproaches\n(0, 0)\n.\n\n## Multiple limits\nAlthough less commonly used, there is another type of limit for a multivariable function, known as the **multiple limit**.\nFor a two-variable function, this is the **double limit**.\nLet be defined on we say **the double limit of f as x approaches p and y approaches q is L**, written\nif the following condition holds:\nFor such a double limit to exist, this definition requires the value of f approaches L along every possible path approaching\n(p, q)\n, excluding the two lines\nx = p\nand\ny = q\n. As a result, the multiple limit is a weaker notion than the ordinary limit: if the ordinary limit exists and equals L, then the multiple limit exists and also equals L. The converse is not true: the existence of the multiple limits does not imply the existence of the ordinary limit.\nConsider the example\nwhere\nbut\ndoes not exist.\nIf the domain of f is restricted to then the two definitions of limits coincide.\n\n## Multiple limits at infinity\nThe concept of multiple limit can extend to the limit at infinity, in a way similar to that of a single variable function.\nFor we say **the double limit of f as x and y approaches infinity is L**, written\nif the following condition holds:\nWe say **the double limit of f as x and y approaches minus infinity is L**, written\nif the following condition holds:\n\n## Pointwise limits and uniform limits\nLet Instead of taking limit as\n(x, y) → (p, q)\n, we may consider taking the limit of just one variable, say,\nx → p\n, to obtain a single-variable function of y, namely In fact, this limiting process can be done in two distinct ways.\nThe first one is called **pointwise limit**.\nWe say **the pointwise limit of f as x approaches p is g**, denoted\nor\nAlternatively, we may say **f tends to g pointwise as x approaches p**, denoted\nor\nThis limit exists if the following holds:\nHere,\nδ = δ(ε, y)\nis a function of both ε and y. Each δ is chosen for a specific point of y. Hence we say the limit is pointwise in y. For example,\nhas a pointwise limit of constant zero function\nbecause for every fixed y, the limit is clearly 0.\nThis argument fails if y is not fixed: if y is very close to\nπ/2\n, the value of the fraction may deviate from 0.\nThis leads to another definition of limit, namely the **uniform limit**.\nWe say **the uniform limit of f on T as x approaches p is g**, denoted\nor\nAlternatively, we may say **f tends to g uniformly on T as x approaches p**, denoted\nor\nThis limit exists if the following holds:\nHere,\nδ = δ(ε)\nis a function of only ε but not y. In other words, δ is uniformly applicable to all y in T. Hence we say the limit is uniform in y. For example,\nhas a uniform limit of constant zero function\nbecause for all real y,\ncos y\nis bounded between\n[−1, 1]\n. Hence no matter how y behaves, we may use the [sandwich theorem](https://bliptext.com/articles/sandwich-theorem) to show that the limit is 0.\n\n## Iterated limits\nLet We may consider taking the limit of just one variable, say,\nx → p\n, to obtain a single-variable function of y, namely and then take limit in the other variable, namely\ny → q\n, to get a number L. Symbolically,\nThis limit is known as **iterated limit** of the multivariable function.\nThe order of taking limits may affect the result, i.e.,\nin general.\nA sufficient condition of equality is given by the [Moore-Osgood theorem](https://bliptext.com/articles/moore-osgood-theorem), which requires the limit to be uniform on T.\n\n# Functions on metric spaces\nSuppose M and N are subsets of [metric spaces](https://bliptext.com/articles/metric-spaces) A and B, respectively, and\nf : M → N\nis defined between M and N, with\nx ∈ M\n, p a [limit point](https://bliptext.com/articles/limit-point) of M and\nL ∈ N\n. It is said that **the limit of f as x approaches p is L** and write\nif the following property holds:\nAgain, 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\nf(p)\nis defined it need not be equal to L.\n\n## Euclidean metric\nThe limit in [Euclidean space](https://bliptext.com/articles/euclidean-space) is a direct generalization of limits to [vector-valued functions](https://bliptext.com/articles/vector-valued-functions).\nFor example, we may consider a function such that\nThen, under the usual [Euclidean metric](https://bliptext.com/articles/euclidean-metric),\nif the following holds:\nIn this example, the function concerned are finite-[dimension](https://bliptext.com/articles/dimension-vector-space) vector-valued function.\nIn this case, the **limit 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:\n\n## Manhattan metric\nOne might also want to consider spaces other than Euclidean space.\nAn example would be the Manhattan space.\nConsider such that\nThen, under the [Manhattan metric](https://bliptext.com/articles/manhattan-metric),\nif the following holds:\nSince this is also a finite-dimension vector-valued function, the limit theorem stated above also applies.\n\n## Uniform metric\nFinally, we will discuss the limit in [function space](https://bliptext.com/articles/function-space), which has infinite dimensions.\nConsider a function\nf(x, y)\nin the function space We want to find out as x approaches p, how\nf(x, y)\nwill tend to another function\ng(y)\n, which is in the function space T \\to \\R.\nThe \"closeness\" in this function space may be measured under the [uniform metric](https://bliptext.com/articles/uniform-metric).\nThen, we will say **the uniform limit of f on T as x approaches p is g** and write\nor\nif the following holds:\nIn fact, one can see that this definition is equivalent to that of the uniform limit of a multivariable function introduced in the previous section.\n\n# Functions on topological spaces\nSuppose X and Y are [topological spaces](https://bliptext.com/articles/topological-space) with Y a [Hausdorff space](https://bliptext.com/articles/hausdorff-space).\nLet p be a [limit point](https://bliptext.com/articles/limit-point) of\nΩ ⊆ X\n, and\nL ∈ Y\n. For a function\nf : Ω → Y\n, it is said that the **limit of f as x approaches p is L**, written\nif the following property holds:\nThis last part of the definition can also be phrased \"there exists an open [punctured neighbourhood](https://bliptext.com/articles/neighbourhood-mathematics) U of p such that\nf(U ∩ Ω) ⊆ V\n\".\nThe 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.\nIn particular, if the domain of f is\nX − {p}\n(or all of X), then the limit of f as\nx → p\nexists and is equal to L if, for all subsets\nΩ\nof X with limit point p, the limit of the restriction of f to\nΩ\nexists 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 limits](https://bliptext.com/articles/one-sided-limit) either fail to exist or do not agree.\nSuch a view is fundamental in the field of [general topology](https://bliptext.com/articles/general-topology), where limits and continuity at a point are defined in terms of special families of subsets, called [filters](https://bliptext.com/articles/filter-set-theory), or generalized sequences known as [nets](https://bliptext.com/articles/net-mathematics).\nAlternatively, 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.\nIn 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.\nA function is continuous at a limit point p of and in its domain if and only if\nf(p)\nis the (or, in the general case, a) limit of\nf(x)\nas x tends to p.\nThere is another type of limit of a function, namely the **sequential limit**.\nLet\nf : X → Y\nbe a mapping from a topological space X into a Hausdorff space Y,\np ∈ X\na limit point of X and\nL ∈ Y\n. The sequential limit of f as x tends to p is L if\nX − {p}\nthat [converges](https://bliptext.com/articles/limit-of-a-sequence) to p, the sequence\nf(xn)\n[converges](https://bliptext.com/articles/limit-of-a-sequence) to L.\nIf 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.\nIf in addition X is [metrizable](https://bliptext.com/articles/metrizable), 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.\n\n# Other characterizations\n\n## In terms of sequences\nFor functions on the real line, one way to define the limit of a function is in terms of the limit of sequences.\n(This definition is usually attributed to [Eduard Heine](https://bliptext.com/articles/eduard-heine).) In this setting:\nif, and only if, for all sequences xn (with xn not equal to a for all n) converging to a the sequence\nf(xn)\nconverges to L. It was shown by [Sierpiński](https://bliptext.com/articles/sierpi-ski) 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](https://bliptext.com/articles/axiom-of-choice).\nNote that defining what it means for a sequence xn to converge to a requires the [epsilon, delta method](https://bliptext.com/articles/epsilon-delta-method).\nSimilarly as it was the case of Weierstrass's definition, a more general Heine definition applies to functions defined on [subsets](https://bliptext.com/articles/subset) of the real line.\nLet f be a real-valued function with the domain\nDm(f )\n. Let a be the limit of a sequence of elements of\nDm(f ) \\ {a}.\nThen the limit (in this sense) of f is L as x approaches p if for every sequence\nxn ∈ Dm(f ) \\ {a}\n(so that for all n, xn is not equal to a) that converges to a, the sequence\nf(xn)\nconverges to L. This is the same as the definition of a sequential limit in the preceding section obtained by regarding the subset\nDm(f )\nof \\R as a metric space with the induced metric.\n\n## In non-standard calculus\nIn non-standard calculus the limit of a function is defined by:\nif and only if for all x\\in \\R^*, f^*(x)-L is infinitesimal whenever\nx − a\nis infinitesimal.\nHere \\R^* are the [hyperreal numbers](https://bliptext.com/articles/hyperreal-number) and f* is the natural extension of f to the non-standard real numbers.\n[Keisler](https://bliptext.com/articles/howard-jerome-keisler) proved that such a hyperreal [definition of limit](https://bliptext.com/articles/non-standard-calculus) reduces the quantifier complexity by two quantifiers.\nOn 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.\nBŀaszczyk et al. detail the usefulness of [microcontinuity](https://bliptext.com/articles/microcontinuity) in developing a transparent definition of uniform continuity, and characterize Hrbacek's criticism as a \"dubious lament\".\n\n## In terms of nearness\nAt the 1908 international congress of mathematics [F. Riesz](https://bliptext.com/articles/frigyes-riesz) introduced an alternate way defining limits and continuity in concept called \"nearness\".\nA point x is defined to be near a set if for every\nr > 0\nthere is a point\na ∈ A\nso that\n. In this setting the\nif and only if for all L is near\nf(A)\nwhenever a is near A. Here\nf(A)\nis the set This definition can also be extended to metric and topological spaces.\n\n# Relationship to continuity\nThe notion of the limit of a function is very closely related to the concept of continuity.\nA function f is said to be [continuous](https://bliptext.com/articles/continuous-function) at c if it is both defined at c and its value at c equals the limit of f as x approaches c:\nWe have here assumed that c is a [limit point](https://bliptext.com/articles/limit-point) of the domain of f.\n\n# Properties\nIf 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.\nThe function f is [continuous](https://bliptext.com/articles/continuous-function) at p if and only if the limit of\nf(x)\nas x approaches p exists and is equal to\nf(p)\n. If\nf : M → N\nis 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\nf(p)\n.\nIf N is a [normed vector space](https://bliptext.com/articles/normed-vector-space), then the limit operation is linear in the following sense: if the limit of\nf(x)\nas x approaches p is L and the limit of\ng(x)\nas x approaches p is P, then the limit of\nf(x) + g(x)\nas x approaches p is\nL + P\n. If a is a scalar from the base [field](https://bliptext.com/articles/field-mathematics), then the limit of\naf(x)\nas x approaches p is aL.\nIf f and g are real-valued (or complex-valued) functions, then taking the limit of an operation on\nf(x)\nand\ng(x)\n(e.g.,\nf + g\n,\nf − g\n,\nf × g\n,\nf / g\n, fg) under certain conditions is compatible with the operation of limits of\nf(x)\nand\ng(x)\n. This fact is often called the **algebraic limit theorem**.\nThe 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).\nAdditionally, 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).\nThese rules are also valid for one-sided limits, including when p is ∞ or −∞.\nIn 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.\n(see also [Extended real number line](https://bliptext.com/articles/extended-real-number-line)).\nIn other cases the limit on the left may still exist, although the right-hand side, called an [indeterminate form](https://bliptext.com/articles/indeterminate-form), does not allow one to determine the result.\nThis depends on the functions f and g. These indeterminate forms are:\nSee further [L'Hôpital's rule](https://bliptext.com/articles/) below and [Indeterminate form](https://bliptext.com/articles/indeterminate-form).\n\n## Limits of compositions of functions\nIn general, from knowing that and it does not follow that\nHowever, this \"chain rule\" does hold if one of the following additional conditions holds:\nf(b) = c\n(that is, f is continuous at b), or\nδ > 0\nsuch that if\n0 <\nthen\n).\nAs an example of this phenomenon, consider the following function that violates both additional restrictions:\nSince the value at\nf(0)\nis a [removable discontinuity](https://bliptext.com/articles/removable-discontinuity),\nfor all a. Thus, the naïve chain rule would suggest that the limit of\nf(f(x))\nis 0.\nHowever, it is the case that\nand so for all a.\n\n## Limits of special interest\n\n### Rational functions\nFor n a nonnegative integer and constants and\nThis can be proven by dividing both the numerator and denominator by xn.\nIf the numerator is a polynomial of higher degree, the limit does not exist.\nIf the denominator is of higher degree, the limit is 0.\n\n### Trigonometric functions\n\n### Exponential functions\n\n### Logarithmic functions\n\n## L'Hôpital's rule\nThis rule uses [derivatives](https://bliptext.com/articles/derivative) to find limits of [indeterminate forms](https://bliptext.com/articles/indeterminate-forms)\n0/0\nor\n±∞/∞\n, and only applies to such cases.\nOther indeterminate forms may be manipulated into this form.\nGiven two functions\nf(x)\nand\ng(x)\n, defined over an [open interval](https://bliptext.com/articles/open-interval) I containing the desired limit point c, then if:\nthen:\nNormally, the first condition is the most important one.\nFor example:\n\n## Summations and integrals\nSpecifying an infinite bound on a summation or integral is a common shorthand for specifying a limit.\nA short way to write the limit\nis An important example of limits of sums such as these are [series](https://bliptext.com/articles/series-mathematics).\nA short way to write the limit\nis\nA short way to write the limit\nis","revisionCount":0},"hasMore":false,"nextCursor":null}