In calculus, the **(ε, δ)-definition of limit** ("epsilon–delta definition of limit") is a formalization of the notion of limit. The concept is due to Augustin-Louis Cauchy, who never gave an (
*Cours d'Analyse*, but occasionally used
*f*(*x*) approaches the value *L* as the variable *x* approaches the value *c* if *f*(*x*) can be made as close as desired to *L* by taking *x* sufficiently close to *c*.

## Contents

## History

Although the Greeks examined limiting process, such as the Babylonian method, they probably had no concept similar to the modern limit. The need for the concept of a limit came into force in the 17th century when Pierre de Fermat attempted to find the slope of the tangent line at a point

The key to the above calculation is that since

This problem reappeared later in the 1600s at the center of the development of calculus because calculations such as Fermat's are important to the calculation of derivatives. Isaac Newton first developed calculus via an infinitesimal quantity called a fluxion. He developed them in reference to the idea of an "infinitely small moment in time..." However, Netwon later rejected fluxions in favor of a theory of ratios that is close to the modern
*not* itself a ratio, as he wrote:

Additionally, Newton occasionally explained limits in terms similar to the epsilon–delta definition. Gottfried Wilhelm Leibniz developed an infinitesimal of his own and tried to provide it with a rigorous footing, but it was still greeted with unease by some mathematicians and philosophers.

Augustin-Louis Cauchy gave a definition of limit in terms of a more primitive notion he called a *variable quantity*. He never gave an epsilon–delta definition of limit (Grabiner 1981). Some of Cauchy's proofs contain indications of the epsilon–delta method. Whether or not his foundational approach can be considered a harbinger of Weierstrass's is a subject of scholarly dispute. Grabiner feels that it is, while Schubring (2005) disagrees. Nakane concludes that Cauchy and Weierstrass gave the same name to different notions of limit.

Eventually, Weierstrass and Bolzano are credited with providing a rigorous footing for calculus in the form of the modern

This is not to say that the limiting definition was free of problems as although it removed the need to use infinitesimals, it did require the construction of the real numbers by Richard Dedekind. This is also not to say that infinitesimals have no place in modern mathematics as later mathematicians were able to rigorously create infinitesimal quantities as part of the hyperreal number or surreal number systems. Moreover, it is possible to rigorously develop calculus with these quantities and they have other mathematical uses.

## Informal statement

A viable intuitive or provisional definition is that a "function *f* approaches the limit *L* near *a* (symbolically,
*f(x)* as close as we like to *L* by requiring that *x* be sufficiently close to, but unequal to, *a*."

When we say that two things are close (such as *f(x)* and *L* or *x* and *a*) we mean that the distance between them is small. When *f(x)*, *L*, *x*, and *a* are real numbers, distance is the absolute value of one quantity minus the other. Thus, when we say *f(x)* is close to *L* we mean
*x* and *a* are close, we mean that

When we say that we can make *f(x)* as close as we like to *L*, we mean that for **all** non-zero distances,
*f(x)* and *L* smaller than

When we say that we can make *f(x)* as close as we like to *L* by requiring that *x* be sufficiently close to, but, unequal to, a, we mean that for all non-zero distances
*x* and *a* is less than
*f(x)* and *L* is smaller than

The aspect that must be grasped is that the definition requires the following conversation. One is provided with any challenge
*f*,*a*, and *L*. One must answer with a

## Precise statement for real valued functions

The

Let

if for every

Symbolically:

If

## Precise statement for functions between metric spaces

The definition can be generalized to functions that map between metric spaces. These spaces come with a function, called a metric, that takes two points in the space and returns a real number that represents the distance between the two points. The generalized definition is as follows:

Suppose

We say that

if for every

Since

## Negation of the precise statement

The negation of the definition is a follows:

Suppose

We say that

if there exists an

We say that

For the negation of a real valued function defined on the real numbers, simply set

## Precise statement for limits at infinity

The precise statement for limits at infinity is as follows:

Suppose

We say that

if for every

## Example 1

We will show that

We let

Since sine is bounded above by 1 and below by -1,

Thus, if we take

## Example 2

Let us prove the statement that

for any real number

Let

We start by factoring:

We recognize that

So we suppose

Thus,

Thus via the triangle inequality,

Thus, if we further suppose that

then

In summary, we set

So, if

Thus, we have found a

for any real number

## Example 3

Let us prove the statement that

This is easily shown through graphical understandings of the limit, and as such serves as a strong basis for introduction to proof. According to the formal definition above, a limit statement is correct if and only if confining

to

Simplifying, factoring, and dividing 3 on the right hand side of the implication yields

which immediately gives the required result if we choose

Thus the proof is completed. The key to the proof lies in the ability of one to choose boundaries in

## 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*:

If the condition 0 < |*x* − *c*| is left out of the definition of limit, then requiring *f*(*x*) to have a limit at *c* would be the same as requiring *f*(*x*) to be continuous at *c*.

*f* is said to be continuous on an interval *I* if it is continuous at every point *c* of *I*.

## Comparison with infinitesimal definition

Keisler proved that a hyperreal definition of limit reduces the quantifier complexity by two quantifiers. Namely,
*L* as
*a* if and only if for every infinitesimal *e*, the value
*L*; see microcontinuity for a related definition of continuity, essentially due to Cauchy. Infinitesimal calculus textbooks based on Robinson's approach provide definitions of continuity, derivative, and integral at standard points in terms of infinitesimals. Once notions such as continuity have been thoroughly explained via the approach using microcontinuity, the epsilon–delta approach is presented as well. Karel Hrbáček argues that the definitions of continuity, derivative, and integration in Robinson-style non-standard analysis must be grounded in the ε–δ method in order to cover also non-standard values of the input. Błaszczyk et al. argue that microcontinuity is useful in developing a transparent definition of uniform continuity, and characterize the criticism by Hrbáček as a "dubious lament". Hrbáček proposes an alternative non-standard analysis, which (unlike Robinson's) has many "levels" of infinitesimals, so that limits at one level can be defined in terms of infinitesimals at the next level.