In mathematics, the limit of a function is a fundamental concept in analysis. Informally, a function f(x) has a limit L at a point p if the value of f(x) can be made as close to L as desired, by making x close enough to p. Formal definitions, first devised in the early 19th century, are given below. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Analysis has its beginnings in the rigorous formulation of calculus. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ...

History

Although implicit in the development of Calculus of the 17th and 18th centuries, the modern notion of the limit of a function goes back to Bolzano who, in 1817, introduced the basics of the epsilon-delta technique (MacTutor History). However, his work was not known during his lifetime. Cauchy discussed limits in his Cours d'analyse (1821) and seems to have expressed the essence of the idea, but not in a systematic fashion (Jeff Miller). The first rigorous public presentation of the technique was given by Weierstrass in the 1850s and 1860s (MacTutor History) and has since become the standard method for dealing with limits. Bernard Bolzano Bernard (Bernhard) Placidus Johann Nepomuk Bolzano (October 5, 1781 – December 18, 1848) was a Bohemian mathematician, theologian, philosopher, logician and antimilitarist of German mother tongue. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ...

The written notation using the lim abbreviation together with the arrow below is due to Hardy in his book A Course of Pure Mathematics in 1908 (Jeff Miller). G. H. Hardy Professor Godfrey Harold Hardy FRS (February 7, 1877 – December 1, 1947) was a prominent English mathematician, known for his achievements in number theory and mathematical analysis. ...

Explanation

Imagine a plane flying over a landscape represented by the graph of y = f(x). Its horizontal position is measured by the value of x, much like the position given by a map of the land or by a global positioning system. Its altitude is given by the coordinate y. It's flying towards the horizontal position given by x = p. As it does so, it notices that its altitude approaches L. If later asked to guess the altitude over x = p, it would then answer L, even if it had never actually reached that position.

What does it mean to say that its altitude approaches L? It means that its 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 the plane: it must get within ten meters of L. The plane reports back that it can get within ten meters of L, since it states that when it is within fifty horizontal meters of p, its altitude is always ten meters or less from L.

We then change our accuracy goal: can it get within one meter? Yes. If it is within seven horizontal meters of p, then its altitude remains within one meter of the target L. In summary, to say that the plane's altitude approaches L as its horizontal position approaches p means that for every target accuracy goal, there is some area of p whose altitude remains within 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.

This explicit statement is quite close to the formal definition of the limit of a function with values in a topological space. In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...

Definitions

The following definitions are the generally accepted ones for the limit of a function in various contexts.

Functions on the real line

Suppose f : R → R is defined on the real line and p,L ∈ R then we say the limit of f as x approaches p is L and write In mathematics, the real line is simply the set of real numbers. ...

if and only if for every real ε > 0 there exists a real δ > 0 such that 0 < | x - p | < δ implies | f(x) - L | < ε. Note particularly that f(p) need not be defined. ↔ ⇔ ≡ logical symbols representing iff. ...

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

or

respectively. If both of these limits are equal to L then this can be referred to as the limit of f(x) at p. Conversely, if they are not both equal to L then the limit, as such, does not exist.

If the limit does not exist there is a non-zero oscillation.

Functions on metric spaces

Suppose f : (M,d_M) → (N,d_N) is defined between two metric spaces, with x ∈ M, p a limit point of M and L ∈ N. We say that the limit of f as x approaches p is L and write In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematics, informally speaking, a limit point (or cluster point) of a set S in a topological space X is a point x in X that can be approximated by points of S other than x as well as one pleases. ...

if and only if for every ε > 0 there exists a δ > 0 such that, d_N(f(x), L) < ε whenever 0 < d_M(x, p) < δ. Again, note that p need not be in the domain of f, nor does L need to be in the range of f. ↔ ⇔ ≡ logical symbols representing iff. ...

An alternative definition using the concept of neighbourhood is as follows: In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ...

if and only if for every neighbourhood V of L in N there exists a neighbourhood U of p in M, such that f(U - {p}) ⊆ V. ↔ ⇔ ≡ logical symbols representing iff. ...

Functions on topological spaces

Suppose X,Y are topological spaces with Y a Hausdorff space. Let p be a limit point of X, and L ∈Y. For a function f : X-{p} → Y, we say that the limit of f as x approaches p is L (i.e., f(x)→L as x→p) and write Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In mathematics, informally speaking, a limit point (or cluster point) of a set S in a topological space X is a point x in X that can be approximated by points of S other than x as well as one pleases. ...

if and only if for every neighborhood V of L, there exists a neighborhood U of p such that f(U- {p}) ⊆ V. In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ...

Note that the domain of f does not need to contain p. If it does, then the value of f at p is irrelevant to the definition of the limit. The last part of the definition can also be phrased "there exists a punctured neighbourhoodhttp://en.wikipedia.org/wiki/Neighbourhood_%28mathematics%29#Punctured_neighbourhood U of p such that f(U) ⊆ V ". In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ...

One can formulate other similar definitions of the limit in a topological space. In one version, the domain of the function f is a subset Ω of the topological space X. In this case, the point p must be a limit point of Ω, and the limit is taken with respect to the induced topology on Ω (one-sided limits, where the limit is taken inside an interval at one of the endpoints, are a special case of this). In mathematics, the domain of a function is the set of all input values to the function. ... In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology (or the relative topology, or the induced topology). ... In mathematics, a one-sided limit is where the limit of a function is defined in moving in the positive or negative direction, but not both. ...

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 and only if for all subsets Ω of X with limit point p the limit of the restriction of f to Ω exists and is equal to L. Sometimes this criterion is used to establish the non-existence of the two-sided limit of a function on R by showing that the one-sided 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. In mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them. ... In mathematics, a filter is a special subset of a partially ordered set. ... In topology and related areas of mathematics a net or Moore-Smith sequence is a generalization of a sequence, intended to unify the various notions of limit and generalize them to arbitrary topological spaces. ...

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 will 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 in a limit point p of and in its domain if and only f(p) is "the" (or in the general case: "a") limit of f(x) as x tends to p.

Limit of a function at infinity

Limit of a function at infinity exists if, for every ε > 0 there exists an S > 0 ; so that | f(x) - L | < ε for all x > S.

If the affinely extended real number system (extended real line) R is considered, i.e., R ∪ {-∞, +∞}, then it is possible to define limits of a function at infinity. Image File history File links Limit-at-infinity-graph. ... Image File history File links Limit-at-infinity-graph. ... In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ (pronounced plus infinity and minus infinity). These new elements are not real numbers. ...

Suppose f(x) is a real-valued function such that x may increase or decrease indefinitely, then we say that the limit of f as x approaches infinity is L and we write

if and only if for every ε > 0 there exists S > 0 such that | f(x) - L | < ε whenever x > S. ↔ ⇔ ≡ logical symbols representing iff. ...

Similarly, we say that the limit of f as x approaches infinity is infinity and we write

if and only if for every R > 0 there exists S > 0 such that for all real numbers f(x) > R whenever x > S. ↔ ⇔ ≡ logical symbols representing iff. ...

In an analogous way, the following expressions can be defined:

.

These notions of a limit attempt to provide a metric space interpretation to limits at infinity. However, note that these notions of a limit are consistent with the topological space definition of limit if

• a neighborhood of -∞ is defined to contain an interval [-∞,c) where c∈R
• a neighborhood of ∞ is defined to contain an interval (c,∞] where c∈R
• a neighborhood of a∈R is defined in the normal way metric space R

In this case, R is a topological space and any function of the form f:X → Y with X,Y⊆ R is subject to the topological definition of a limit. Note that with this topological definition, it is easy to define infinite limits at finite points, which have not been defined above in the metric sense. In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...

Evaluating limits at infinity for rational functions

There are three basic rules for evaluating limits at infinity for a rational function f(x) = p(x)/q(x):

• 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 x = L. Polynomials do not have horizontal asymptotes; they may occur with rational functions.

Complex-valued functions

The complex plane with metric d(x,y): = | x − y | is also a metric space. There are two different types of limits when we consider complex-valued functions. In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ...

Limit of a function at a point

Suppose f is a complex-valued function, then we write

if and only if

for every ε > 0 there exists a δ >0 such that for all real numbers x with 0<|x-p|<δ, we have |f(x)-L|<ε

It is just a particular case of functions over metric spaces with both M and N are the complex plane.

Limit of a function of more than one variable

By noting that |x-p| represents a distance, the definition of a limit can be extended to functions of more than one variable. In the case of a function f : R² → R,

if and only if ↔ ⇔ ≡ logical symbols representing iff. ...

for every ε > 0 there exists a δ > 0 such that for all (x,y) with 0 < ||(x,y)-(p,q)|| < δ, we have |f(x,y)-L| < ε

where ||(x,y)-(p,q)|| represents the Euclidean distance. This can be extended to any number of variables. In mathematics, the Euclidean distance or Euclidean metric is the ordinary distance between two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. ...

Properties

To say that the limit of a function f at p is L is equivalent to saying

for every convergent sequence (x_n) in M with limit equal to p, the sequence (f(x_n)) converges with limit L.

If the sets A, B, ... form a finite partition of the function domain, , ... and the relative limit for each of those sets exist and is the equal to, say, L, then the limit exists for the point x and is equal to L. The limit of a sequence is one of the oldest concepts in mathematical analysis. ...

The function f is continuous at p if and only if the limit of f(x) as x approaches p exists and is finite. Equivalently, f transforms every sequence in M which converges towards p into a sequence in N which converges towards f(p). In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

Again, 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. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

Taking the limit of functions is compatible with the algebraic operations, provided the limits on the right sides of the identity below exist:

(the last provided that the denominator is non-zero). In each case above, when the limits on the right do not exist, or, in the last case, when the limits in both the numerator and the denominator are zero, nonetheless the limit on the left may still exist -- this depends on which functions f and g are.

These rules are also valid for one-sided limits, for the case p = ±∞, and also for infinite limits using the rules

• q + ∞ = ∞ for q ≠ -∞
• q × ∞ = ∞ if q > 0
• q × ∞ = −∞ if q < 0
• q / ∞ = 0 if q ≠ ± ∞

(see extended real number line). The extended real number line is obtained from the real number line R by adding two elements: +&#8734; and &#8722;&#8734; (which are not considered to be real numbers). ...

Note that there is no general rule for the case q / 0; it all depends on the way 0 is approached. Indeterminate forms — for instance, 0/0, 0×∞, ∞−∞, and ∞/∞ — are also not covered by these rules, but the corresponding limits can often be determined with L'Hôpital's rule or the Squeeze theorem. In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression whose limit cannot be evaluated by substituting the limits of the subexpressions. ... In calculus, lHôpitals rule uses derivatives to help compute limits with indeterminate forms. ... In calculus, the squeeze theorem (also known as the pinching theorem or the sandwich theorem, sometimes the squeeze lemma) is a theorem regarding the limit of a function. ...

See also

• One-sided limit
• Limit of a sequence
• Net (topology)
• Big O notation
• Limit superior and limit inferior
• l'Hôpital's rule
• Squeeze theorem

In mathematics, a one-sided limit is where the limit of a function is defined in moving in the positive or negative direction, but not both. ... The limit of a sequence is one of the oldest concepts in mathematical analysis. ... In mathematics the term net has at least two meanings. ... For other uses, see Big O. In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithms usage of computational resources (usually running time or memory). ... In mathematics, the limit inferior and limit superior (also called infimum limit and supremum limit) of a sequence can be thought of as limiting bounds on the sequence. ... In calculus, lHôpitals rule uses derivatives to help compute limits with indeterminate forms. ... In calculus, the squeeze theorem (also known as the pinching theorem or the sandwich theorem, sometimes the squeeze lemma) is a theorem regarding the limit of a function. ...

References

• Visual Calculus by Lawrence S. Husch, University of Tennessee (2001)

• Apostol, Tom M., Mathematical Analysis, 2nd ed. Addison-Wesley, 1974. ISBN 0201002884.

• Sutherland, W. A., Introduction to Metric and Topological Spaces. Oxford University Press, Oxford, 1975. ISBN 0 19 853161 3.