(ε, δ)-definition of limit

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Mar 16, 2025 / 01:19

In mathematics the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that

(ε, δ)-definition of limit
(ε, δ)-definition of limit
(ε, δ)-definition of limit

In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input which may or may not be in the domain of the function.

1 0.841471...
0.1 0.998334...
0.01 0.999983...

Although the function is not defined at zero, as x becomes closer and closer to zero, becomes arbitrarily close to 1. In other words, the limit of as x approaches zero, equals 1.

Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f(x) to every input x. We say that the function has a limit L at an input p, if f(x) gets closer and closer to L as x moves closer and closer to p. More specifically, 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.

The notion of a limit has many applications in modern calculus. In particular, the many definitions of continuity employ the concept of limit: roughly, a function is continuous if all of its limits agree with the values of the function. The concept of limit also appears in the definition of the derivative: in the calculus of one variable, this is the limiting value of the slope of secant lines to the graph of a function.

History

Although implicit in the development of calculus of the 17th and 18th centuries, the modern idea of the limit of a function goes back to Bernard 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. Bruce Pourciau argues that Isaac Newton, in his 1687 Principia, demonstrates a more sophisticated understanding of limits than he is generally given credit for, including being the first to present an epsilon argument.

In his 1821 book Cours d'analyse, Augustin-Louis Cauchy discussed variable quantities, infinitesimals and limits, and defined continuity of image 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. In 1861, Karl Weierstrass first introduced the epsilon-delta definition of limit in the form it is usually written today. He also introduced the notations image and image

The modern notation of placing the arrow below the limit symbol is due to G. H. Hardy, which is introduced in his book A Course of Pure Mathematics in 1908.

Motivation

Imagine a person walking on a landscape represented by the graph y = f(x). Their horizontal position is given by x, much like the position given by a map of the land or by a global positioning system. Their altitude is given by the coordinate y. Suppose they walk towards a position x = p, 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 x = p, they would reply by saying y = L.

What, then, does it mean to say, their altitude is approaching L? It means that their altitude gets nearer and nearer to L—except for a possible small error in accuracy. For example, suppose we set a particular accuracy goal for our traveler: they must get within ten meters of L. They report back that indeed, they can get within ten vertical meters of L, arguing that as long as they are within fifty horizontal meters of p, their altitude is always within ten meters of L.

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 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 correspond to all the horizontal positions, except maybe the horizontal position p itself, in that neighbourhood fulfill that accuracy goal.

The initial informal statement can now be explicated:

The limit of a function f(x) as x approaches p is a number L with the following property: given any target distance from L, there is a distance from p within which the values of f(x) remain within the target distance.

In fact, this explicit statement is quite close to the formal definition of the limit of a function, with values in a topological space.

More specifically, to say that

image

is to say that f(x) can be made as close to L as desired, by making x close enough, but not equal, to p.

The following definitions, known as (ε, δ)-definitions, are the generally accepted definitions for the limit of a function in various contexts.

Functions of a single variable

(ε, δ)-definition of limit

image
For the depicted f, a, and b, we can ensure that the value f(x) is within an arbitrarily small interval (b – ε, b + ε) by restricting x to a sufficiently small interval (a – δ, a + δ). Hence f(x) → b as xa.

Suppose image is a function defined on the real line, and there are two real numbers p and L. One would say. The limit of f of x, as x approaches p, exists, and it equals L and write,

image

or alternatively, say f(x) tends to L as x tends to p, and write,

image

if the following property holds: for every real ε > 0, there exists a real δ > 0 such that for all real x, 0 < |xp| < δ implies |f(x) − L| < ε. Symbolically, image

For example, we may say image because for every real ε > 0, we can take δ = ε/4, so that for all real x, if 0 < |x − 2| < δ, then |4x + 1 − 9| < ε.

A more general definition applies for functions defined on subsets of the real line. Let S be a subset of image Let image be a real-valued function. Let p be a point such that there exists some open interval (a, b) containing p with image It is then said that the limit of f as x approaches p is L, if:

For every real ε > 0, there exists a real δ > 0 such that for all x ∈ (a, b), 0 < |xp| < δ implies that |f(x) − L| < ε.

Or, symbolically: image

For example, we may say image because for every real ε > 0, we can take δ = ε, so that for all real x ≥ −3, if 0 < |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 on f being defined at p, nor on the value f(p)—if it is defined. For example, let image image because for every ε > 0, we can take δ = ε/2, so that for all real x ≠ 1, if 0 < |x − 1| < δ, then |f(x) − 3| < ε. Note that here f(1) is undefined.

In fact, a limit can exist in image which equals image where int S is the interior of S, and iso Sc are the isolated points of the complement of S. In our previous example where image image image We see, specifically, this definition of limit allows a limit to exist at 1, but not 0 or 2.

The letters ε and δ can be understood as "error" and "distance". In fact, Cauchy used ε as an abbreviation for "error" in some of his work, though in his definition of continuity, he used an infinitesimal image rather than either ε or δ (see Cours 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 that δ and ε represent distances helps suggest these generalizations.

Existence and one-sided limits

image
The limit as image differs from that as image Therefore, the limit as xx0 does not exist.

Alternatively, x may approach p from above (right) or below (left), in which case the limits may be written as

image

or

image

image
The first three functions have points for which the limit does not exist, while the functionimageis not defined at image, but its limit does exist.

respectively. If these limits exist at p and are equal there, then this can be referred to as the limit of f(x) at p. If the one-sided limits exist at p, but are unequal, then there is no limit at p (i.e., the limit at p does not exist). If either one-sided limit does not exist at p, then the limit at p also does not exist.

A formal definition is as follows. The limit of f as x approaches p from above is L if:

For every ε > 0, there exists a δ > 0 such that whenever 0 < xp < δ, we have |f(x) − L| < ε.

image

The limit of f as x approaches p from below is L if:

For every ε > 0, there exists a δ > 0 such that whenever 0 < px < δ, we have |f(x) − L| < ε.

image

If the limit does not exist, then the oscillation of f at p is non-zero.

More general definition using limit points and subsets

Limits can also be defined by approaching from subsets of the domain.

In general: Let image be a real-valued function defined on some image Let p be a limit point of some image—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 image if the following holds:

For every ε > 0, there exists a δ > 0 such that for all xT, 0 < |xp| < δ implies that |f(x) − L| < ε.

image

Note, 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 (–∞, a)), and right-handed limits (e.g., by taking T 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 the square root function image can have limit 0 as x approaches 0 from above: image since for every ε > 0, we may take δ = ε2 such that for all x ≥ 0, if 0 < |x − 0| < δ, then |f(x) − 0| < ε.

This 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.

Notably, the previous two-sided definition works on image which is a subset of the limit points of S.

For example, let image The previous two-sided definition would work at image but it wouldn't work at 0 or 2, which are limit points of S.

Deleted versus non-deleted limits

The 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 image be a real-valued function. The non-deleted limit of f, as x approaches p, is L if

For every ε > 0, there exists a δ > 0 such that for all xS, |xp| < δ implies |f(x) − L| < ε.

image

The definition is the same, except that the neighborhood |xp| < δ now includes the point p, in contrast to the deleted neighborhood 0 < |xp| < δ. This makes the definition of a non-deleted limit less general. One of the advantages of working with non-deleted limits is that they allow to state the theorem about limits of compositions without any constraints on the functions (other than the existence of their non-deleted limits).

Bartle notes that although by "limit" some authors do mean this non-deleted limit, deleted limits are the most popular.

Examples

Non-existence of one-sided limit(s)

image
Function without a limit at an essential discontinuity

The function image has no limit at x0 = 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 other x-coordinate.

The function image (a.k.a., the Dirichlet function) has no limit at any x-coordinate.

Non-equality of one-sided limits

The function image has a limit at every non-zero x-coordinate (the limit equals 1 for negative x and equals 2 for positive x). The limit at x = 0 does not exist (the left-hand limit equals 1, whereas the right-hand limit equals 2).

Limits at only one point

The functions image and image both have a limit at x = 0 and it equals 0.

Limits at countably many points

The function image has a limit at any x-coordinate of the form image where n is any integer.

Limits involving infinity

Limits at infinity

image
The limit of this function at infinity exists

Let image be a function defined on image The limit of f as x approaches infinity is L, denoted

image

means that:

For every ε > 0, there exists a c > 0 such that whenever +x > c, we have |f(x) − L| < ε.

image

Similarly, the limit of f as x approaches minus infinity is L, denoted

image

means that:

For every ε > 0, there exists a c > 0 such that whenever x < −c, we have |f(x) − L| < ε.

image

For example, image because for every ε > 0, we can take c = 3/ε such that for all real x, if x > c, then |f(x) − 4| < ε.

Another example is that image because for every ε > 0, we can take c = max{1, −ln(ε)} such that for all real x, if x < −c, then |f(x) − 0| < ε.

Infinite limits

For a function whose values grow without bound, the function diverges and the usual limit does not exist. However, in this case one may introduce limits with infinite values.

Let image be a function defined on image The statement the limit of f as x approaches p is infinity, denoted

image

means that:

For every N > 0, there exists a δ > 0 such that whenever 0 < |xp| < δ, we have f(x) > N.

image

The statement the limit of f as x approaches p is minus infinity, denoted

image

means that:

For every N > 0, there exists a δ > 0 such that whenever 0 < |xp| < δ, we have f(x) < −N.

image

For example, image because for every N > 0, we can take image such that for all real x > 0, if 0 < x − 1 < δ, then f(x) > N.

These ideas can be used together to produce definitions for different combinations, such as

image or image

For example, image because for every N > 0, we can take δ = eN such that for all real x > 0, if 0 < x − 0 < δ, then f(x) < −N.

Limits involving infinity are connected with the concept of asymptotes.

These notions of a limit attempt to provide a metric space interpretation to limits at infinity. In fact, they are consistent with the topological space definition of limit if

  • a neighborhood of −∞ is defined to contain an interval [−∞, c) for some image
  • a neighborhood of ∞ is defined to contain an interval (c, ∞] where image and
  • a neighborhood of image is defined in the normal way metric space image

In this case, image is a topological space and any function of the form image with image is subject to the topological definition of a limit. Note that with this topological definition, it is easy to define infinite limits at finite points, which have not been defined above in the metric sense.

Alternative notation

Many authors allow for the projectively extended real line to be used as a way to include infinite values as well as extended real line. With this notation, the extended real line is given as image and the projectively extended real line is image where a neighborhood of ∞ is a set of the form image 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, image does not possess a central limit (which is normal):

image

In contrast, when working with the projective real line, infinities (much like 0) are unsigned, so, the central limit does exist in that context:

image

In fact there are a plethora of conflicting formal systems in use. In certain applications of numerical differentiation and integration, it is, for example, convenient to have signed zeroes. A simple reason has to do with the converse of image namely, it is convenient for image to be considered true. Such zeroes can be seen as an approximation to infinitesimals.

Limits at infinity for rational functions

image
Horizontal asymptote about y = 4

There are three basic rules for evaluating limits at infinity for a rational function image (where p and q are polynomials):

  • If the degree of p is greater than the degree of q, then the limit is positive or negative infinity depending on the signs of the leading coefficients;
  • If the degree of p and q are equal, the limit is the leading coefficient of p divided by the leading coefficient of q;
  • If the degree of p is less than the degree of q, the limit is 0.

If the limit at infinity exists, it represents a horizontal asymptote at y = L. Polynomials do not have horizontal asymptotes; such asymptotes may however occur with rational functions.

Functions of more than one variable

Ordinary limits

By noting that |xp| represents a distance, the definition of a limit can be extended to functions of more than one variable. In the case of a function image defined on image we defined the limit as follows: the limit of f as (x, y) approaches (p, q) is L, written

image

if the following condition holds:

For every ε > 0, there exists a δ > 0 such that for all x in S and y in T, whenever image we have |f(x, y) − L| < ε,

or formally: image

Here image is the Euclidean distance between (x, y) and (p, q). (This can in fact be replaced by any norm ||(x, y) − (p, q)||, and be extended to any number of variables.)

For example, we may say image because for every ε > 0, we can take image such that for all real x ≠ 0 and real y ≠ 0, if image then |f(x, y) − 0| < ε.

Similar to the case in single variable, the value of f at (p, q) does not matter in this definition of limit.

For such a multivariable limit to exist, this definition requires the value of f approaches L along every possible path approaching (p, q). In the above example, the function image satisfies this condition. This can be seen by considering the polar coordinates image which gives image Here θ = θ(r) is a function of r which controls the shape of the path along which f is approaching (p, q). Since cos θ is bounded between [−1, 1], by the sandwich theorem, this limit tends to 0.

In contrast, the function image does not have a limit at (0, 0). Taking the path (x, y) = (t, 0) → (0, 0), we obtain image while taking the path (x, y) = (t, t) → (0, 0), we obtain image

Since the two values do not agree, f does not tend to a single value as (x, y) approaches (0, 0).

Multiple limits

Although less commonly used, there is another type of limit for a multivariable function, known as the multiple limit. For a two-variable function, this is the double limit. Let image be defined on image we say the double limit of f as x approaches p and y approaches q is L, written

image

if the following condition holds:

For every ε > 0, there exists a δ > 0 such that for all x in S and y in T, whenever 0 < |xp| < δ and 0 < |yq| < δ, we have |f(x, y) − L| < ε.

image

For such a double limit to exist, this definition requires the value of f approaches L along every possible path approaching (p, q), excluding the two lines x = p and y = q. 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. Consider the example image where image but image does not exist.

If the domain of f is restricted to image then the two definitions of limits coincide.

Multiple limits at infinity

The concept of multiple limit can extend to the limit at infinity, in a way similar to that of a single variable function. For image we say the double limit of f as x and y approaches infinity is L, written image

if the following condition holds:

For every ε > 0, there exists a c > 0 such that for all x in S and y in T, whenever x > c and y > c, we have |f(x, y) − L| < ε.

image

We say the double limit of f as x and y approaches minus infinity is L, written image

if the following condition holds:

For every ε > 0, there exists a c > 0 such that x in S and y in T, whenever x < −c and y < −c, we have |f(x, y) − L| < ε.

image

Pointwise limits and uniform limits

Let image Instead of taking limit as (x, y) → (p, q), we may consider taking the limit of just one variable, say, xp, to obtain a single-variable function of y, namely image In fact, this limiting process can be done in two distinct ways. The first one is called pointwise limit. We say the pointwise limit of f as x approaches p is g, denoted image or image

Alternatively, we may say f tends to g pointwise as x approaches p, denoted image or image

This limit exists if the following holds:

For every ε > 0 and every fixed y in T, there exists a δ(ε, y) > 0 such that for all x in S, whenever 0 < |xp| < δ, we have |f(x, y) − g(y)| < ε.

image

Here, δ = δ(ε, y) is 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, image has a pointwise limit of constant zero function image because for every fixed y, the limit is clearly 0. This argument fails if y is not fixed: if y is very close to π/2, the value of the fraction may deviate from 0.

This leads to another definition of limit, namely the uniform limit. We say the uniform limit of f on T as x approaches p is g, denoted image or image

Alternatively, we may say f tends to g uniformly on T as x approaches p, denoted image or image

This limit exists if the following holds:

For every ε > 0, there exists a δ(ε) > 0 such that for all x in S and y in T, whenever 0 < |xp| < δ, we have |f(x, y) − g(y)| < ε.

image

Here, δ = δ(ε) is 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, image has a uniform limit of constant zero function image because for all real y, cos y is bounded between [−1, 1]. Hence no matter how y behaves, we may use the sandwich theorem to show that the limit is 0.

Iterated limits

Let image We may consider taking the limit of just one variable, say, xp, to obtain a single-variable function of y, namely image and then take limit in the other variable, namely yq, to get a number L. Symbolically, image

This limit is known as iterated limit of the multivariable function. The order of taking limits may affect the result, i.e.,

image in general.

A sufficient condition of equality is given by the Moore-Osgood theorem, which requires the limit image to be uniform on T.

Functions on metric spaces

Suppose M and N are subsets of metric spaces A and B, respectively, and f : MN is defined between M and N, with xM, p a limit point of M and LN. It is said that the limit of f as x approaches p is L and write

image

if the following property holds:

For every ε > 0, there exists a δ > 0 such that for all points xM, 0 < dA(x, p) < δ implies dB(f(x), L) < ε.

image

Again, note that p need not be in the domain of f, nor does L need to be in the range of f, and even if f(p) is defined it need not be equal to L.

Euclidean metric

The limit in Euclidean space is a direct generalization of limits to vector-valued functions. For example, we may consider a function image such that image Then, under the usual Euclidean metric, image if the following holds:

For every ε > 0, there exists a δ > 0 such that for all x in S and y in T, image implies image

image

In this example, the function concerned are finite-dimension vector-valued function. In 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:image

Manhattan metric

One might also want to consider spaces other than Euclidean space. An example would be the Manhattan space. Consider image such that image Then, under the Manhattan metric, image if the following holds:

For every ε > 0, there exists a δ > 0 such that for all x in S, 0 < |xp| < δ implies |f1L1| + |f2L2| < ε.

image

Since this is also a finite-dimension vector-valued function, the limit theorem stated above also applies.

Uniform metric

Finally, we will discuss the limit in function space, which has infinite dimensions. Consider a function f(x, y) in the function space image We want to find out as x approaches p, how f(x, y) will tend to another function g(y), which is in the function space image The "closeness" in this function space may be measured under the uniform metric. Then, we will say the uniform limit of f on T as x approaches p is g and write image or image

if the following holds:

For every ε > 0, there exists a δ > 0 such that for all x in S, 0 < |xp| < δ implies image

image

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.

Functions on topological spaces

Suppose X and Y are topological spaces with Y a Hausdorff space. Let p be a limit point of Ω ⊆ X, and LY. For a function f : Ω → Y, it is said that the limit of f as x approaches p is L, written

image

if the following property holds:

For every open neighborhood V of L, there exists an open neighborhood U of p such that f(U ∩ Ω − {p}) ⊆ V.

This last part of the definition can also be phrased "there exists an open punctured neighbourhood U of p such that f(U ∩ Ω) ⊆ V".

The domain of f does not need to contain p. If it does, then the value of f at p is irrelevant to the definition of the limit. In particular, if the domain of f is X − {p} (or all of X), then the limit of f as xp exists and is equal to L if, for all subsets Ω of X with limit point p, the limit of the restriction of f to Ω exists and is equal to L. Sometimes this criterion is used to establish the non-existence of the two-sided limit of a function on image by showing that the one-sided limits either fail to exist or do not agree. Such a view is fundamental in the field of general topology, where limits and continuity at a point are defined in terms of special families of subsets, called filters, or generalized sequences known as nets.

Alternatively, the requirement that Y be a Hausdorff space can be relaxed to the assumption that Y be a general topological space, but then the limit of a function may not be unique. In particular, one can no longer talk about the limit of a function at a point, but rather a limit or the set of limits at a point.

A function is continuous at a limit point p of and in its domain if and only if f(p) is the (or, in the general case, a) limit of f(x) as x tends to p.

There is another type of limit of a function, namely the sequential limit. Let f : XY be a mapping from a topological space X into a Hausdorff space Y, pX a limit point of X and LY. The sequential limit of f as x tends to p is L if

For every sequence (xn) in X − {p} that converges to p, the sequence f(xn) converges to L.

If L is the limit (in the sense above) of f as x approaches p, then it is a sequential limit as well, however the converse need not hold in general. If in addition X is metrizable, then L is the sequential limit of f as x approaches p if and only if it is the limit (in the sense above) of f as x approaches p.

Other characterizations

In terms of sequences

For functions on the real line, one way to define the limit of a function is in terms of the limit of sequences. (This definition is usually attributed to Eduard Heine.) In this setting: image if, and only if, for all sequences xn (with, for all n, xn not equal to a) converging to a the sequence f(xn) converges to L. It was shown by Sierpiński in 1916 that proving the equivalence of this definition and the definition above, requires and is equivalent to a weak form of the axiom of choice. Note that defining what it means for a sequence xn to converge to a requires the epsilon, delta method.

Similarly as it was the case of Weierstrass's definition, a more general Heine definition applies to functions defined on subsets of the real line. Let f be a real-valued function with the domain Dm(f ). Let a be the limit of a sequence of elements of Dm(f ) \ {a}. Then the limit (in this sense) of f is L as x approaches a if for every sequence xnDm(f ) \ {a} (so that for all n, xn is not equal to a) that converges to a, the sequence f(xn) converges to L. This is the same as the definition of a sequential limit in the preceding section obtained by regarding the subset Dm(f ) of image as a metric space with the induced metric.

In non-standard calculus

In non-standard calculus the limit of a function is defined by: image if and only if for all image image is infinitesimal whenever xa is infinitesimal. Here image are the hyperreal numbers and f* is the natural extension of f to the non-standard real numbers. Keisler proved that such a hyperreal definition of limit reduces the quantifier complexity by two quantifiers. On the other hand, Hrbacek writes that for the definitions to be valid for all hyperreal numbers they must implicitly be grounded in the ε-δ method, and claims that, from the pedagogical point of view, the hope that non-standard calculus could be done without ε-δ methods cannot be realized in full. Bŀaszczyk et al. detail the usefulness of microcontinuity in developing a transparent definition of uniform continuity, and characterize Hrbacek's criticism as a "dubious lament".

In terms of nearness

At the 1908 international congress of mathematics F. Riesz introduced an alternate way defining limits and continuity in concept called "nearness". A point x is defined to be near a set image if for every r > 0 there is a point aA so that |xa| < r. In this setting the image if and only if for all image L is near f(A) whenever a is near A. Here f(A) is the set image This definition can also be extended to metric and topological spaces.

Relationship to continuity

The notion of the limit of a function is very closely related to the concept of continuity. A function f is said to be continuous at c if it is both defined at c and its value at c equals the limit of f as x approaches c:

image We have here assumed that c is a limit point of the domain of f.

Properties

If a function f is real-valued, then the limit of f at p is L if and only if both the right-handed limit and left-handed limit of f at p exist and are equal to L.

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

If N is a normed vector space, then the limit operation is linear in the following sense: if the limit of f(x) as x approaches p is L and the limit of g(x) as x approaches p is P, then the limit of f(x) + g(x) as x approaches p is L + P. If a is a scalar from the base field, then the limit of af(x) as x approaches p is aL.

If f and g are real-valued (or complex-valued) functions, then taking the limit of an operation on f(x) and g(x) (e.g., f + g, fg, f × g, f / g, f g) under certain conditions is compatible with the operation of limits of f(x) and g(x). This fact is often called the algebraic limit theorem. The main condition needed to apply the following rules is that the limits on the right-hand sides of the equations exist (in other words, these limits are finite values including 0). Additionally, the identity for division requires that the denominator on the right-hand side is non-zero (division by 0 is not defined), and the identity for exponentiation requires that the base is positive, or zero while the exponent is positive (finite).

image

These rules are also valid for one-sided limits, including when p is ∞ or −∞. In each rule above, when one of the limits on the right is ∞ or −∞, the limit on the left may sometimes still be determined by the following rules.

image

(see also Extended real number line).

In other cases the limit on the left may still exist, although the right-hand side, called an indeterminate form, does not allow one to determine the result. This depends on the functions f and g. These indeterminate forms are:

image

See further L'Hôpital's rule below and Indeterminate form.

Limits of compositions of functions

In general, from knowing that image and image it does not follow that image However, this "chain rule" does hold if one of the following additional conditions holds:

  • f(b) = c (that is, f is continuous at b), or
  • g does not take the value b near a (that is, there exists a δ > 0 such that if 0 < |xa| < δ then |g(x) − b| > 0).

As an example of this phenomenon, consider the following function that violates both additional restrictions:

image

Since the value at f(0) is a removable discontinuity, image for all a. Thus, the naïve chain rule would suggest that the limit of f(f(x)) is 0. However, it is the case that image and so image for all a.

Limits of special interest

Rational functions

For n a nonnegative integer and constants image and image

image

This can be proven by dividing both the numerator and denominator by xn. If the numerator is a polynomial of higher degree, the limit does not exist. If the denominator is of higher degree, the limit is 0.

Trigonometric functions

image

Exponential functions

image

Logarithmic functions

image

L'Hôpital's rule

This rule uses derivatives to find limits of indeterminate forms 0/0 or ±∞/∞, and only applies to such cases. Other indeterminate forms may be manipulated into this form. Given two functions f(x) and g(x), defined over an open interval I containing the desired limit point c, then if:

  1. image or image and
  2. image and image are differentiable over image and
  3. image for all image and
  4. image exists,

then:

In mathematics the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input which may or may not be in the domain of the function x displaystyle x sin xx displaystyle frac sin x x 1 0 841471 0 1 0 998334 0 01 0 999983 Although the function sin xx displaystyle tfrac sin x x is not defined at zero as x becomes closer and closer to zero sin xx displaystyle tfrac sin x x becomes arbitrarily close to 1 In other words the limit of sin xx displaystyle tfrac sin x x as x approaches zero equals 1 Formal definitions first devised in the early 19th century are given below Informally a function f assigns an output f x to every input x We say that the function has a limit L at an input p if f x gets closer and closer to L as x moves closer and closer to p More specifically 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 The notion of a limit has many applications in modern calculus In particular the many definitions of continuity employ the concept of limit roughly a function is continuous if all of its limits agree with the values of the function The concept of limit also appears in the definition of the derivative in the calculus of one variable this is the limiting value of the slope of secant lines to the graph of a function HistoryAlthough implicit in the development of calculus of the 17th and 18th centuries the modern idea of the limit of a function goes back to Bernard Bolzano who in 1817 introduced the basics of the epsilon delta technique see e d definition of limit below to define continuous functions However his work was not known during his lifetime Bruce Pourciau argues that Isaac Newton in his 1687 Principia demonstrates a more sophisticated understanding of limits than he is generally given credit for including being the first to present an epsilon argument In his 1821 book Cours d analyse Augustin Louis Cauchy discussed variable quantities infinitesimals and limits and defined continuity of y f x displaystyle 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 In 1861 Karl Weierstrass first introduced the epsilon delta definition of limit in the form it is usually written today He also introduced the notations lim textstyle lim and limx x0 textstyle textstyle lim x to x 0 displaystyle The modern notation of placing the arrow below the limit symbol is due to G H Hardy which is introduced in his book A Course of Pure Mathematics in 1908 MotivationImagine a person walking on a landscape represented by the graph y f x Their horizontal position is given by x much like the position given by a map of the land or by a global positioning system Their altitude is given by the coordinate y Suppose they walk towards a position x p 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 x p they would reply by saying y L What then does it mean to say their altitude is approaching L It means that their altitude gets nearer and nearer to L except for a possible small error in accuracy For example suppose we set a particular accuracy goal for our traveler they must get within ten meters of L They report back that indeed they can get within ten vertical meters of L arguing that as long as they are within fifty horizontal meters of p their altitude is always within ten meters of L 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 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 correspond to all the horizontal positions except maybe the horizontal position p itself in that neighbourhood fulfill that accuracy goal The initial informal statement can now be explicated The limit of a function f x as x approaches p is a number L with the following property given any target distance from L there is a distance from p within which the values of f x remain within the target distance In fact this explicit statement is quite close to the formal definition of the limit of a function with values in a topological space More specifically to say that limx pf x L displaystyle lim x to p f x L is to say that f x can be made as close to L as desired by making x close enough but not equal to p The following definitions known as e d definitions are the generally accepted definitions for the limit of a function in various contexts Functions of a single variable e d definition of limit For the depicted f a and b we can ensure that the value f x is within an arbitrarily small interval b e b e by restricting x to a sufficiently small interval a d a d Hence f x b as x a Suppose f R R displaystyle f mathbb R rightarrow mathbb R is a function defined on the real line and there are two real numbers p and L One would say The limit of f of x as x approaches p exists and it equals L and write limx pf x L displaystyle lim x to p f x L or alternatively say f x tends to L as x tends to p and write f x L as x p displaystyle f x to L text as x to p if the following property holds for every real e gt 0 there exists a real d gt 0 such that for all real x 0 lt x p lt d implies f x L lt e Symbolically e gt 0 d gt 0 x R 0 lt x p lt d f x L lt e displaystyle forall varepsilon gt 0 exists delta gt 0 forall x in mathbb R 0 lt x p lt delta implies f x L lt varepsilon For example we may say limx 2 4x 1 9 displaystyle lim x to 2 4x 1 9 because for every real e gt 0 we can take d e 4 so that for all real x if 0 lt x 2 lt d then 4x 1 9 lt e A more general definition applies for functions defined on subsets of the real line Let S be a subset of R displaystyle mathbb R Let f S R displaystyle f S to mathbb R be a real valued function Let p be a point such that there exists some open interval a b containing p with a p p b S displaystyle a p cup p b subset S It is then said that the limit of f as x approaches p is L if For every real e gt 0 there exists a real d gt 0 such that for all x a b 0 lt x p lt d implies that f x L lt e Or symbolically e gt 0 d gt 0 x a b 0 lt x p lt d f x L lt e displaystyle forall varepsilon gt 0 exists delta gt 0 forall x in a b 0 lt x p lt delta implies f x L lt varepsilon For example we may say limx 1x 3 2 displaystyle lim x to 1 sqrt x 3 2 because for every real e gt 0 we can take d e so that for all real x 3 if 0 lt x 1 lt d then f x 2 lt e 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 on f being defined at p nor on the value f p if it is defined For example let f 0 1 1 2 R f x 2x2 x 1x 1 displaystyle f 0 1 cup 1 2 to mathbb R f x tfrac 2x 2 x 1 x 1 limx 1f x 3 displaystyle lim x to 1 f x 3 because for every e gt 0 we can take d e 2 so that for all real x 1 if 0 lt x 1 lt d then f x 3 lt e Note that here f 1 is undefined In fact a limit can exist in x R a b Rp a b and a p p b S displaystyle x in mathbb R exists a b subset mathbb R quad p in a b text and a p cup p b subset S which equals int S iso Sc displaystyle operatorname int S cup operatorname iso S c where int S is the interior of S and iso Sc are the isolated points of the complement of S In our previous example where S 0 1 1 2 displaystyle S 0 1 cup 1 2 int S 0 1 1 2 displaystyle operatorname int S 0 1 cup 1 2 iso Sc 1 displaystyle operatorname iso S c 1 We see specifically this definition of limit allows a limit to exist at 1 but not 0 or 2 The letters e and d can be understood as error and distance In fact Cauchy used e as an abbreviation for error in some of his work though in his definition of continuity he used an infinitesimal a displaystyle alpha rather than either e or d see Cours d Analyse In these terms the error e in the measurement of the value at the limit can be made as small as desired by reducing the distance d to the limit point As discussed below this definition also works for functions in a more general context The idea that d and e represent distances helps suggest these generalizations Existence and one sided limits The limit as x x0 displaystyle x to x 0 differs from that as x x0 displaystyle x to x 0 Therefore the limit as x x0 does not exist Alternatively x may approach p from above right or below left in which case the limits may be written as limx p f x L displaystyle lim x to p f x L or limx p f x L displaystyle lim x to p f x L The first three functions have points for which the limit does not exist while the functionf x sin x x displaystyle f x frac sin x x is not defined at x 0 displaystyle x 0 but its limit does exist respectively If these limits exist at p and are equal there then this can be referred to as the limit of f x at p If the one sided limits exist at p but are unequal then there is no limit at p i e the limit at p does not exist If either one sided limit does not exist at p then the limit at p also does not exist A formal definition is as follows The limit of f as x approaches p from above is L if For every e gt 0 there exists a d gt 0 such that whenever 0 lt x p lt d we have f x L lt e e gt 0 d gt 0 x a b 0 lt x p lt d f x L lt e displaystyle forall varepsilon gt 0 exists delta gt 0 forall x in a b 0 lt x p lt delta implies f x L lt varepsilon The limit of f as x approaches p from below is L if For every e gt 0 there exists a d gt 0 such that whenever 0 lt p x lt d we have f x L lt e e gt 0 d gt 0 x a b 0 lt p x lt d f x L lt e displaystyle forall varepsilon gt 0 exists delta gt 0 forall x in a b 0 lt p x lt delta implies f x L lt varepsilon If the limit does not exist then the oscillation of f at p is non zero More general definition using limit points and subsets Limits can also be defined by approaching from subsets of the domain In general Let f S R displaystyle f S to mathbb R be a real valued function defined on some S R displaystyle S subseteq mathbb R Let p be a limit point of some T S displaystyle 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 limx px Tf x L displaystyle lim x to p atop x in T f x L if the following holds For every e gt 0 there exists a d gt 0 such that for all x T 0 lt x p lt d implies that f x L lt e e gt 0 d gt 0 x T 0 lt x p lt d f x L lt e displaystyle forall varepsilon gt 0 exists delta gt 0 forall x in T 0 lt x p lt delta implies f x L lt varepsilon Note 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 a and right handed limits e g by taking T 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 the square root function f x x displaystyle f x sqrt x can have limit 0 as x approaches 0 from above limx 0x 0 x 0 displaystyle lim x to 0 atop x in 0 infty sqrt x 0 since for every e gt 0 we may take d e2 such that for all x 0 if 0 lt x 0 lt d then f x 0 lt e This 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 Notably the previous two sided definition works on int S iso Sc displaystyle operatorname int S cup operatorname iso S c which is a subset of the limit points of S For example let S 0 1 1 2 displaystyle S 0 1 cup 1 2 The previous two sided definition would work at 1 iso Sc 1 displaystyle 1 in operatorname iso S c 1 but it wouldn t work at 0 or 2 which are limit points of S Deleted versus non deleted limits The 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 f S R displaystyle f S to mathbb R be a real valued function The non deleted limit of f as x approaches p is L if For every e gt 0 there exists a d gt 0 such that for all x S x p lt d implies f x L lt e e gt 0 d gt 0 x S x p lt d f x L lt e displaystyle forall varepsilon gt 0 exists delta gt 0 forall x in S x p lt delta implies f x L lt varepsilon The definition is the same except that the neighborhood x p lt d now includes the point p in contrast to the deleted neighborhood 0 lt x p lt d This makes the definition of a non deleted limit less general One of the advantages of working with non deleted limits is that they allow to state the theorem about limits of compositions without any constraints on the functions other than the existence of their non deleted limits Bartle notes that although by limit some authors do mean this non deleted limit deleted limits are the most popular Examples Non existence of one sided limit s Function without a limit at an essential discontinuity The function f x sin 5x 1 for x lt 10 for x 1110x 10 for x gt 1 displaystyle f x begin cases sin frac 5 x 1 amp text for x lt 1 0 amp text for x 1 2pt frac 1 10x 10 amp text for x gt 1 end cases has no limit at x0 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 other x coordinate The function f x 1x rational 0x irrational displaystyle f x begin cases 1 amp x text rational 0 amp x text irrational end cases a k a the Dirichlet function has no limit at any x coordinate Non equality of one sided limits The function f x 1 for x lt 02 for x 0 displaystyle f x begin cases 1 amp text for x lt 0 2 amp text for x geq 0 end cases has a limit at every non zero x coordinate the limit equals 1 for negative x and equals 2 for positive x The limit at x 0 does not exist the left hand limit equals 1 whereas the right hand limit equals 2 Limits at only one point The functions f x xx rational 0x irrational displaystyle f x begin cases x amp x text rational 0 amp x text irrational end cases and f x x x rational 0x irrational displaystyle f x begin cases x amp x text rational 0 amp x text irrational end cases both have a limit at x 0 and it equals 0 Limits at countably many points The function f x sin xx irrational 1x rational displaystyle f x begin cases sin x amp x text irrational 1 amp x text rational end cases has a limit at any x coordinate of the form p2 2np displaystyle tfrac pi 2 2n pi where n is any integer Limits involving infinityLimits at infinity The limit of this function at infinity exists Let f S R displaystyle f S to mathbb R be a function defined on S R displaystyle S subseteq mathbb R The limit of f as x approaches infinity is L denoted limx f x L displaystyle lim x to infty f x L means that For every e gt 0 there exists a c gt 0 such that whenever x gt c we have f x L lt e e gt 0 c gt 0 x S x gt c f x L lt e displaystyle forall varepsilon gt 0 exists c gt 0 forall x in S x gt c implies f x L lt varepsilon Similarly the limit of f as x approaches minus infinity is L denoted limx f x L displaystyle lim x to infty f x L means that For every e gt 0 there exists a c gt 0 such that whenever x lt c we have f x L lt e e gt 0 c gt 0 x S x lt c f x L lt e displaystyle forall varepsilon gt 0 exists c gt 0 forall x in S x lt c implies f x L lt varepsilon For example limx 3sin xx 4 4 displaystyle lim x to infty left frac 3 sin x x 4 right 4 because for every e gt 0 we can take c 3 e such that for all real x if x gt c then f x 4 lt e Another example is that limx ex 0 displaystyle lim x to infty e x 0 because for every e gt 0 we can take c max 1 ln e such that for all real x if x lt c then f x 0 lt e Infinite limits For a function whose values grow without bound the function diverges and the usual limit does not exist However in this case one may introduce limits with infinite values Let f S R displaystyle f S to mathbb R be a function defined on S R displaystyle S subseteq mathbb R The statement the limit of f as x approaches p is infinity denoted limx pf x displaystyle lim x to p f x infty means that For every N gt 0 there exists a d gt 0 such that whenever 0 lt x p lt d we have f x gt N N gt 0 d gt 0 x S 0 lt x p lt d f x gt N displaystyle forall N gt 0 exists delta gt 0 forall x in S 0 lt x p lt delta implies f x gt N The statement the limit of f as x approaches p is minus infinity denoted limx pf x displaystyle lim x to p f x infty means that For every N gt 0 there exists a d gt 0 such that whenever 0 lt x p lt d we have f x lt N N gt 0 d gt 0 x S 0 lt x p lt d f x lt N displaystyle forall N gt 0 exists delta gt 0 forall x in S 0 lt x p lt delta implies f x lt N For example limx 11 x 1 2 displaystyle lim x to 1 frac 1 x 1 2 infty because for every N gt 0 we can take d 1Nd 1N textstyle delta tfrac 1 sqrt N delta tfrac 1 sqrt N such that for all real x gt 0 if 0 lt x 1 lt d then f x gt N These ideas can be used together to produce definitions for different combinations such as limx f x displaystyle lim x to infty f x infty or limx p f x displaystyle lim x to p f x infty For example limx 0 ln x displaystyle lim x to 0 ln x infty because for every N gt 0 we can take d e N such that for all real x gt 0 if 0 lt x 0 lt d then f x lt N Limits involving infinity are connected with the concept of asymptotes These notions of a limit attempt to provide a metric space interpretation to limits at infinity In fact they are consistent with the topological space definition of limit if a neighborhood of is defined to contain an interval c for some c R displaystyle c in mathbb R a neighborhood of is defined to contain an interval c where c R displaystyle c in mathbb R and a neighborhood of a R displaystyle a in mathbb R is defined in the normal way metric space R displaystyle mathbb R In this case R displaystyle overline mathbb R is a topological space and any function of the form f X Y displaystyle f X to Y with X Y R displaystyle X Y subseteq overline mathbb R is subject to the topological definition of a limit Note that with this topological definition it is easy to define infinite limits at finite points which have not been defined above in the metric sense Alternative notation Many authors allow for the projectively extended real line to be used as a way to include infinite values as well as extended real line With this notation the extended real line is given as R displaystyle mathbb R cup infty infty and the projectively extended real line is R displaystyle mathbb R cup infty where a neighborhood of is a set of the form x x gt c displaystyle x x gt c The advantage is that one only needs three definitions for limits left right and central to cover all the cases As presented above for a completely rigorous account we would need to consider 15 separate cases for each combination of infinities five directions left central right and three bounds finite or There are also noteworthy pitfalls For example when working with the extended real line x 1 displaystyle x 1 does not possess a central limit which is normal limx 0 1x limx 0 1x displaystyle lim x to 0 1 over x infty quad lim x to 0 1 over x infty In contrast when working with the projective real line infinities much like 0 are unsigned so the central limit does exist in that context limx 0 1x limx 0 1x limx 01x displaystyle lim x to 0 1 over x lim x to 0 1 over x lim x to 0 1 over x infty In fact there are a plethora of conflicting formal systems in use In certain applications of numerical differentiation and integration it is for example convenient to have signed zeroes A simple reason has to do with the converse of limx 0 x 1 displaystyle lim x to 0 x 1 infty namely it is convenient for limx x 1 0 displaystyle lim x to infty x 1 0 to be considered true Such zeroes can be seen as an approximation to infinitesimals Limits at infinity for rational functions Horizontal asymptote about y 4 There are three basic rules for evaluating limits at infinity for a rational function f x p x q x displaystyle f x tfrac p x q x where p and q are polynomials If the degree of p is greater than the degree of q then the limit is positive or negative infinity depending on the signs of the leading coefficients If the degree of p and q are equal the limit is the leading coefficient of p divided by the leading coefficient of q If the degree of p is less than the degree of q the limit is 0 If the limit at infinity exists it represents a horizontal asymptote at y L Polynomials do not have horizontal asymptotes such asymptotes may however occur with rational functions Functions of more than one variableOrdinary limits By noting that x p represents a distance the definition of a limit can be extended to functions of more than one variable In the case of a function f S T R displaystyle f S times T to mathbb R defined on S T R2 displaystyle S times T subseteq mathbb R 2 we defined the limit as follows the limit of f as x y approaches p q is L written lim x y p q f x y L displaystyle lim x y to p q f x y L if the following condition holds For every e gt 0 there exists a d gt 0 such that for all x in S and y in T whenever 0 lt x p 2 y q 2 lt d textstyle 0 lt sqrt x p 2 y q 2 lt delta we have f x y L lt e or formally e gt 0 d gt 0 x S y T 0 lt x p 2 y q 2 lt d f x y L lt e displaystyle forall varepsilon gt 0 exists delta gt 0 forall x in S forall y in T 0 lt sqrt x p 2 y q 2 lt delta implies f x y L lt varepsilon Here x p 2 y q 2 textstyle sqrt x p 2 y q 2 is the Euclidean distance between x y and p q This can in fact be replaced by any norm x y p q and be extended to any number of variables For example we may say lim x y 0 0 x4x2 y2 0 displaystyle lim x y to 0 0 frac x 4 x 2 y 2 0 because for every e gt 0 we can take d e textstyle delta sqrt varepsilon such that for all real x 0 and real y 0 if 0 lt x 0 2 y 0 2 lt d textstyle 0 lt sqrt x 0 2 y 0 2 lt delta then f x y 0 lt e Similar to the case in single variable the value of f at p q does not matter in this definition of limit For such a multivariable limit to exist this definition requires the value of f approaches L along every possible path approaching p q In the above example the function f x y x4x2 y2 displaystyle f x y frac x 4 x 2 y 2 satisfies this condition This can be seen by considering the polar coordinates x y rcos 8 rsin 8 0 0 displaystyle x y r cos theta r sin theta to 0 0 which gives limr 0f rcos 8 rsin 8 limr 0r4cos4 8r2 limr 0r2cos4 8 displaystyle lim r to 0 f r cos theta r sin theta lim r to 0 frac r 4 cos 4 theta r 2 lim r to 0 r 2 cos 4 theta Here 8 8 r is a function of r which controls the shape of the path along which f is approaching p q Since cos 8 is bounded between 1 1 by the sandwich theorem this limit tends to 0 In contrast the function f x y xyx2 y2 displaystyle f x y frac xy x 2 y 2 does not have a limit at 0 0 Taking the path x y t 0 0 0 we obtain limt 0f t 0 limt 00t2 0 displaystyle lim t to 0 f t 0 lim t to 0 frac 0 t 2 0 while taking the path x y t t 0 0 we obtain limt 0f t t limt 0t2t2 t2 12 displaystyle lim t to 0 f t t lim t to 0 frac t 2 t 2 t 2 frac 1 2 Since the two values do not agree f does not tend to a single value as x y approaches 0 0 Multiple limits Although less commonly used there is another type of limit for a multivariable function known as the multiple limit For a two variable function this is the double limit Let f S T R displaystyle f S times T to mathbb R be defined on S T R2 displaystyle S times T subseteq mathbb R 2 we say the double limit of f as x approaches p and y approaches q is L written limx py qf x y L displaystyle lim x to p atop y to q f x y L if the following condition holds For every e gt 0 there exists a d gt 0 such that for all x in S and y in T whenever 0 lt x p lt d and 0 lt y q lt d we have f x y L lt e e gt 0 d gt 0 x S y T 0 lt x p lt d 0 lt y q lt d f x y L lt e displaystyle forall varepsilon gt 0 exists delta gt 0 forall x in S forall y in T 0 lt x p lt delta land 0 lt y q lt delta implies f x y L lt varepsilon For such a double limit to exist this definition requires the value of f approaches L along every possible path approaching p q excluding the two lines x p and y q 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 Consider the example f x y 1forxy 00forxy 0 displaystyle f x y begin cases 1 quad text for quad xy neq 0 0 quad text for quad xy 0 end cases where limx 0y 0f x y 1 displaystyle lim x to 0 atop y to 0 f x y 1 but lim x y 0 0 f x y displaystyle lim x y to 0 0 f x y does not exist If the domain of f is restricted to S p T q displaystyle S setminus p times T setminus q then the two definitions of limits coincide Multiple limits at infinity The concept of multiple limit can extend to the limit at infinity in a way similar to that of a single variable function For f S T R displaystyle f S times T to mathbb R we say the double limit of f as x and y approaches infinity is L written limx y f x y L displaystyle lim x to infty atop y to infty f x y L if the following condition holds For every e gt 0 there exists a c gt 0 such that for all x in S and y in T whenever x gt c and y gt c we have f x y L lt e e gt 0 c gt 0 x S y T x gt c y gt c f x y L lt e displaystyle forall varepsilon gt 0 exists c gt 0 forall x in S forall y in T x gt c land y gt c implies f x y L lt varepsilon We say the double limit of f as x and y approaches minus infinity is L written limx y f x y L displaystyle lim x to infty atop y to infty f x y L if the following condition holds For every e gt 0 there exists a c gt 0 such that x in S and y in T whenever x lt c and y lt c we have f x y L lt e e gt 0 c gt 0 x S y T x lt c y lt c f x y L lt e displaystyle forall varepsilon gt 0 exists c gt 0 forall x in S forall y in T x lt c land y lt c implies f x y L lt varepsilon Pointwise limits and uniform limits Let f S T R displaystyle f S times T to mathbb R 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 of y namely g T R displaystyle g T to mathbb R In fact this limiting process can be done in two distinct ways The first one is called pointwise limit We say the pointwise limit of f as x approaches p is g denoted limx pf x y g y displaystyle lim x to p f x y g y or limx pf x y g y pointwise displaystyle lim x to p f x y g y text pointwise Alternatively we may say f tends to g pointwise as x approaches p denoted f x y g y asx p displaystyle f x y to g y text as x to p or f x y g y pointwiseasx p displaystyle f x y to g y text pointwise text as x to p This limit exists if the following holds For every e gt 0 and every fixed y in T there exists a d e y gt 0 such that for all x in S whenever 0 lt x p lt d we have f x y g y lt e e gt 0 y T d gt 0 x S 0 lt x p lt d f x y g y lt e displaystyle forall varepsilon gt 0 forall y in T exists delta gt 0 forall x in S 0 lt x p lt delta implies f x y g y lt varepsilon Here d d e y is a function of both e and y Each d is chosen for a specific point of y Hence we say the limit is pointwise in y For example f x y xcos y displaystyle f x y frac x cos y has a pointwise limit of constant zero function limx 0f x y 0 y pointwise displaystyle lim x to 0 f x y 0 y text pointwise because for every fixed y the limit is clearly 0 This argument fails if y is not fixed if y is very close to p 2 the value of the fraction may deviate from 0 This leads to another definition of limit namely the uniform limit We say the uniform limit of f on T as x approaches p is g denoted uniflimx py Tf x y g y displaystyle underset x to p atop y in T mathrm unif lim f x y g y or limx pf x y g y uniformly onT displaystyle lim x to p f x y g y text uniformly on T Alternatively we may say f tends to g uniformly on T as x approaches p denoted f x y g y onTasx p displaystyle f x y rightrightarrows g y text on T text as x to p or f x y g y uniformly onTasx p displaystyle f x y to g y text uniformly on T text as x to p This limit exists if the following holds For every e gt 0 there exists a d e gt 0 such that for all x in S and y in T whenever 0 lt x p lt d we have f x y g y lt e e gt 0 d gt 0 x S y T 0 lt x p lt d f x y g y lt e displaystyle forall varepsilon gt 0 exists delta gt 0 forall x in S forall y in T 0 lt x p lt delta implies f x y g y lt varepsilon Here d d e is a function of only e but not y In other words d is uniformly applicable to all y in T Hence we say the limit is uniform in y For example f x y xcos y displaystyle f x y x cos y has a uniform limit of constant zero function limx 0f x y 0 y uniformly onR displaystyle lim x to 0 f x y 0 y text uniformly on mathbb R because for all real y cos y is bounded between 1 1 Hence no matter how y behaves we may use the sandwich theorem to show that the limit is 0 Iterated limits Let f S T R displaystyle f S times T to mathbb R We may consider taking the limit of just one variable say x p to obtain a single variable function of y namely g T R displaystyle g T to mathbb R and then take limit in the other variable namely y q to get a number L Symbolically limy qlimx pf x y limy qg y L displaystyle lim y to q lim x to p f x y lim y to q g y L This limit is known as iterated limit of the multivariable function The order of taking limits may affect the result i e limy qlimx pf x y limx plimy qf x y displaystyle lim y to q lim x to p f x y neq lim x to p lim y to q f x y in general A sufficient condition of equality is given by the Moore Osgood theorem which requires the limit limx pf x y g y displaystyle lim x to p f x y g y to be uniform on T Functions on metric spacesSuppose M and N are subsets of metric spaces A and B respectively and f M N is defined between M and N with x M p a limit point of M and L N It is said that the limit of f as x approaches p is L and write limx pf x L displaystyle lim x to p f x L if the following property holds For every e gt 0 there exists a d gt 0 such that for all points x M 0 lt dA x p lt d implies dB f x L lt e e gt 0 d gt 0 x M 0 lt dA x p lt d dB f x L lt e displaystyle forall varepsilon gt 0 exists delta gt 0 forall x in M 0 lt d A x p lt delta implies d B f x L lt varepsilon Again note that p need not be in the domain of f nor does L need to be in the range of f and even if f p is defined it need not be equal to L Euclidean metric The limit in Euclidean space is a direct generalization of limits to vector valued functions For example we may consider a function f S T R3 displaystyle f S times T to mathbb R 3 such that f x y f1 x y f2 x y f3 x y displaystyle f x y f 1 x y f 2 x y f 3 x y Then under the usual Euclidean metric lim x y p q f x y L1 L2 L3 displaystyle lim x y to p q f x y L 1 L 2 L 3 if the following holds For every e gt 0 there exists a d gt 0 such that for all x in S and y in T 0 lt x p 2 y q 2 lt d textstyle 0 lt sqrt x p 2 y q 2 lt delta implies f1 L1 2 f2 L2 2 f3 L3 2 lt e textstyle sqrt f 1 L 1 2 f 2 L 2 2 f 3 L 3 2 lt varepsilon e gt 0 d gt 0 x S y T 0 lt x p 2 y q 2 lt d f1 L1 2 f2 L2 2 f3 L3 2 lt e displaystyle forall varepsilon gt 0 exists delta gt 0 forall x in S forall y in T left 0 lt sqrt x p 2 y q 2 lt delta implies sqrt f 1 L 1 2 f 2 L 2 2 f 3 L 3 2 lt varepsilon right In this example the function concerned are finite dimension vector valued function In 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 lim x y p q f1 x y f2 x y f3 x y lim x y p q f1 x y lim x y p q f2 x y lim x y p q f3 x y displaystyle lim x y to p q Bigl f 1 x y f 2 x y f 3 x y Bigr left lim x y to p q f 1 x y lim x y to p q f 2 x y lim x y to p q f 3 x y right Manhattan metric One might also want to consider spaces other than Euclidean space An example would be the Manhattan space Consider f S R2 displaystyle f S to mathbb R 2 such that f x f1 x f2 x displaystyle f x f 1 x f 2 x Then under the Manhattan metric limx pf x L1 L2 displaystyle lim x to p f x L 1 L 2 if the following holds For every e gt 0 there exists a d gt 0 such that for all x in S 0 lt x p lt d implies f1 L1 f2 L2 lt e e gt 0 d gt 0 x S 0 lt x p lt d f1 L1 f2 L2 lt e displaystyle forall varepsilon gt 0 exists delta gt 0 forall x in S 0 lt x p lt delta implies f 1 L 1 f 2 L 2 lt varepsilon Since this is also a finite dimension vector valued function the limit theorem stated above also applies Uniform metric Finally we will discuss the limit in function space which has infinite dimensions Consider a function f x y in the function space S T R displaystyle S times T to mathbb R We want to find out as x approaches p how f x y will tend to another function g y which is in the function space T R displaystyle T to mathbb R The closeness in this function space may be measured under the uniform metric Then we will say the uniform limit of f on T as x approaches p is g and write uniflimx py Tf x y g y displaystyle underset x to p atop y in T mathrm unif lim f x y g y or limx pf x y g y uniformly onT displaystyle lim x to p f x y g y text uniformly on T if the following holds For every e gt 0 there exists a d gt 0 such that for all x in S 0 lt x p lt d implies supy T f x y g y lt e displaystyle sup y in T f x y g y lt varepsilon e gt 0 d gt 0 x S 0 lt x p lt d supy T f x y g y lt e displaystyle forall varepsilon gt 0 exists delta gt 0 forall x in S 0 lt x p lt delta implies sup y in T f x y g y lt varepsilon 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 Functions on topological spacesSuppose X and Y are topological spaces with Y a Hausdorff space Let p be a limit point of W X and L Y For a function f W Y it is said that the limit of f as x approaches p is L written limx pf x L displaystyle lim x to p f x L if the following property holds For every open neighborhood V of L there exists an open neighborhood U of p such that f U W p V This last part of the definition can also be phrased there exists an open punctured neighbourhood U of p such that f U W V The domain of f does not need to contain p If it does then the value of f at p is irrelevant to the definition of the limit In particular if the domain of f is X p or all of X then the limit of f as x p exists and is equal to L if for all subsets W of X with limit point p the limit of the restriction of f to W exists and is equal to L Sometimes this criterion is used to establish the non existence of the two sided limit of a function on R displaystyle mathbb R by showing that the one sided limits either fail to exist or do not agree Such a view is fundamental in the field of general topology where limits and continuity at a point are defined in terms of special families of subsets called filters or generalized sequences known as nets Alternatively the requirement that Y be a Hausdorff space can be relaxed to the assumption that Y be a general topological space but then the limit of a function may not be unique In particular one can no longer talk about the limit of a function at a point but rather a limit or the set of limits at a point A function is continuous at a limit point p of and in its domain if and only if f p is the or in the general case a limit of f x as x tends to p There is another type of limit of a function namely the sequential limit Let f X Y be a mapping from a topological space X into a Hausdorff space Y p X a limit point of X and L Y The sequential limit of f as x tends to p is L if For every sequence xn in X p that converges to p the sequence f xn converges to L If L is the limit in the sense above of f as x approaches p then it is a sequential limit as well however the converse need not hold in general If in addition X is metrizable then L is the sequential limit of f as x approaches p if and only if it is the limit in the sense above of f as x approaches p Other characterizationsIn terms of sequences For functions on the real line one way to define the limit of a function is in terms of the limit of sequences This definition is usually attributed to Eduard Heine In this setting limx af x L displaystyle lim x to a f x L if and only if for all sequences xn with for all n xn not equal to a converging to a the sequence f xn converges to L It was shown by Sierpinski 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 Note that defining what it means for a sequence xn to converge to a requires the epsilon delta method Similarly as it was the case of Weierstrass s definition a more general Heine definition applies to functions defined on subsets of the real line Let f be a real valued function with the domain Dm f Let a be the limit of a sequence of elements of Dm f a Then the limit in this sense of f is L as x approaches a if for every sequence xn Dm f a so that for all n xn is not equal to a that converges to a the sequence f xn converges to L This is the same as the definition of a sequential limit in the preceding section obtained by regarding the subset Dm f of R displaystyle mathbb R as a metric space with the induced metric In non standard calculus In non standard calculus the limit of a function is defined by limx af x L displaystyle lim x to a f x L if and only if for all x R displaystyle x in mathbb R f x L displaystyle f x L is infinitesimal whenever x a is infinitesimal Here R displaystyle mathbb R are the hyperreal numbers and f is the natural extension of f to the non standard real numbers Keisler proved that such a hyperreal definition of limit reduces the quantifier complexity by two quantifiers On the other hand Hrbacek writes that for the definitions to be valid for all hyperreal numbers they must implicitly be grounded in the e d method and claims that from the pedagogical point of view the hope that non standard calculus could be done without e d methods cannot be realized in full Bŀaszczyk et al detail the usefulness of microcontinuity in developing a transparent definition of uniform continuity and characterize Hrbacek s criticism as a dubious lament In terms of nearness At the 1908 international congress of mathematics F Riesz introduced an alternate way defining limits and continuity in concept called nearness A point x is defined to be near a set A R displaystyle A subseteq mathbb R if for every r gt 0 there is a point a A so that x a lt r In this setting the limx af x L displaystyle lim x to a f x L if and only if for all A R displaystyle A subseteq mathbb R L is near f A whenever a is near A Here f A is the set f x x A displaystyle f x x in A This definition can also be extended to metric and topological spaces Relationship to continuityThe notion of the limit of a function is very closely related to the concept of continuity A function f is said to be continuous at c if it is both defined at c and its value at c equals the limit of f as x approaches c limx cf x f c displaystyle lim x to c f x f c We have here assumed that c is a limit point of the domain of f PropertiesIf a function f is real valued then the limit of f at p is L if and only if both the right handed limit and left handed limit of f at p exist and are equal to L The function f is continuous at p if and only if the limit of f x as x approaches p exists and is equal to f p If f M N is a function between metric spaces M and N then it is equivalent that f transforms every sequence in M which converges towards p into a sequence in N which converges towards f p If N is a normed vector space then the limit operation is linear in the following sense if the limit of f x as x approaches p is L and the limit of g x as x approaches p is P then the limit of f x g x as x approaches p is L P If a is a scalar from the base field then the limit of af x as x approaches p is aL If f and g are real valued or complex valued functions then taking the limit of an operation on f x and g x e g f g f g f g f g fg under certain conditions is compatible with the operation of limits of f x and g x This fact is often called the algebraic limit theorem The main condition needed to apply the following rules is that the limits on the right hand sides of the equations exist in other words these limits are finite values including 0 Additionally the identity for division requires that the denominator on the right hand side is non zero division by 0 is not defined and the identity for exponentiation requires that the base is positive or zero while the exponent is positive finite limx p f x g x limx pf x limx pg x limx p f x g x limx pf x limx pg x limx p f x g x limx pf x limx pg x limx p f x g x limx pf x limx pg x limx pf x g x limx pf x limx pg x displaystyle begin array lcl displaystyle lim x to p f x g x amp amp displaystyle lim x to p f x lim x to p g x displaystyle lim x to p f x g x amp amp displaystyle lim x to p f x lim x to p g x displaystyle lim x to p f x cdot g x amp amp displaystyle lim x to p f x cdot lim x to p g x displaystyle lim x to p f x g x amp amp displaystyle lim x to p f x lim x to p g x displaystyle lim x to p f x g x amp amp displaystyle lim x to p f x lim x to p g x end array These rules are also valid for one sided limits including when p is or In each rule above when one of the limits on the right is or the limit on the left may sometimes still be determined by the following rules q if q q if q gt 0 if q lt 0q 0 if q and q q 0if q lt 0 if q gt 0q 0if 0 lt q lt 1 if q gt 1q if 0 lt q lt 10if q gt 1 displaystyle begin array rcl q infty amp amp infty text if q neq infty 8pt q times infty amp amp begin cases infty amp text if q gt 0 infty amp text if q lt 0 end cases 6pt displaystyle frac q infty amp amp 0 text if q neq infty text and q neq infty 6pt infty q amp amp begin cases 0 amp text if q lt 0 infty amp text if q gt 0 end cases 4pt q infty amp amp begin cases 0 amp text if 0 lt q lt 1 infty amp text if q gt 1 end cases 4pt q infty amp amp begin cases infty amp text if 0 lt q lt 1 0 amp text if q gt 1 end cases end array see also Extended real number line In other cases the limit on the left may still exist although the right hand side called an indeterminate form does not allow one to determine the result This depends on the functions f and g These indeterminate forms are 00 0 00 01 displaystyle begin array cc displaystyle frac 0 0 amp displaystyle frac pm infty pm infty 6pt 0 times pm infty amp infty infty 8pt qquad 0 0 qquad amp qquad infty 0 qquad 8pt 1 pm infty end array See further L Hopital s rule below and Indeterminate form Limits of compositions of functions In general from knowing that limy bf y c displaystyle lim y to b f y c and limx ag x b displaystyle lim x to a g x b it does not follow that limx af g x c displaystyle lim x to a f g x c However this chain rule does hold if one of the following additional conditions holds f b c that is f is continuous at b or g does not take the value b near a that is there exists a d gt 0 such that if 0 lt x a lt d then g x b gt 0 As an example of this phenomenon consider the following function that violates both additional restrictions f x g x 0if x 01if x 0 displaystyle f x g x begin cases 0 amp text if x neq 0 1 amp text if x 0 end cases Since the value at f 0 is a removable discontinuity limx af x 0 displaystyle lim x to a f x 0 for all a Thus the naive chain rule would suggest that the limit of f f x is 0 However it is the case that f f x 1if x 00if x 0 displaystyle f f x begin cases 1 amp text if x neq 0 0 amp text if x 0 end cases and so limx af f x 1 displaystyle lim x to a f f x 1 for all a Limits of special interest Rational functions For n a nonnegative integer and constants a1 a2 a3 an displaystyle a 1 a 2 a 3 ldots a n and b1 b2 b3 bn displaystyle b 1 b 2 b 3 ldots b n limx a1xn a2xn 1 a3xn 2 anb1xn b2xn 1 b3xn 2 bn a1b1 displaystyle lim x to infty frac a 1 x n a 2 x n 1 a 3 x n 2 dots a n b 1 x n b 2 x n 1 b 3 x n 2 dots b n frac a 1 b 1 This can be proven by dividing both the numerator and denominator by xn If the numerator is a polynomial of higher degree the limit does not exist If the denominator is of higher degree the limit is 0 Trigonometric functions limx 0sin xx 1limx 01 cos xx 0 displaystyle begin array lcl displaystyle lim x to 0 frac sin x x amp amp 1 4pt displaystyle lim x to 0 frac 1 cos x x amp amp 0 end array Exponential functions limx 0 1 x 1x limr 1 1r r elimx 0ex 1x 1limx 0eax 1bx ablimx 0cax 1bx abln climx 0 xx 1 displaystyle begin array lcl displaystyle lim x to 0 1 x frac 1 x amp amp displaystyle lim r to infty left 1 frac 1 r right r e 4pt displaystyle lim x to 0 frac e x 1 x amp amp 1 4pt displaystyle lim x to 0 frac e ax 1 bx amp amp displaystyle frac a b 4pt displaystyle lim x to 0 frac c ax 1 bx amp amp displaystyle frac a b ln c 4pt displaystyle lim x to 0 x x amp amp 1 end array Logarithmic functions limx 0ln 1 x x 1limx 0ln 1 ax bx ablimx 0logc 1 ax bx abln c displaystyle begin array lcl displaystyle lim x to 0 frac ln 1 x x amp amp 1 4pt displaystyle lim x to 0 frac ln 1 ax bx amp amp displaystyle frac a b 4pt displaystyle lim x to 0 frac log c 1 ax bx amp amp displaystyle frac a b ln c end array L Hopital s rule This rule uses derivatives to find limits of indeterminate forms 0 0 or and only applies to such cases Other indeterminate forms may be manipulated into this form Given two functions f x and g x defined over an open interval I containing the desired limit point c then if limx cf x limx cg x 0 displaystyle lim x to c f x lim x to c g x 0 or limx cf x limx cg x displaystyle lim x to c f x pm lim x to c g x pm infty and f displaystyle f and g displaystyle g are differentiable over I c displaystyle I setminus c and g x 0 displaystyle g x neq 0 for all x I c displaystyle x in I setminus c and limx cf x g x displaystyle lim x to c tfrac f x g x exists then limx cf x g x limx cf x g x displaystyle lim x to c frac f x g x lim x to c frac f x g x

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Tuesday, 18 March, 2025
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