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Calculus |
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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.

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, when *f* is applied to any input *sufficiently* close to *p*, the output value is forced *arbitrarily* close to *L*. 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.

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 Bolzano who, in 1817, introduced the basics of the epsilon-delta technique to define continuous functions. However, his work was not known during his lifetime.^{[1]}

In his 1821 book *Cours d'analyse*, Cauchy discussed variable quantities, infinitesimals and limits, and defined continuity of by saying that an infinitesimal change in *x* necessarily produces an infinitesimal change in *y*, while (Grabiner 1983) claims that he used a rigorous epsilon-delta definition in proofs.^{[2]} In 1861, Weierstrass first introduced the epsilon-delta definition of limit in the form it is usually written today.^{[3]} He also introduced the notations **lim** and **lim**_{x→x0}.^{[4]}

The modern notation of placing the arrow below the limit symbol is due to Hardy, which is introduced in his book *A Course of Pure Mathematics* in 1908.^{[5]}

Imagine a person walking over a landscape represented by the graph of *y* = *f*(*x*). Their 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. Their altitude is given by the coordinate *y*. They walk toward the horizontal position given by *x* = *p*. As they get closer and closer to it, they notice that their altitude approaches *L*. If asked about the altitude of *x* = *p*, they would then answer *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*, since they note that when they are within fifty horizontal meters of *p*, their altitude is *always* ten meters or less from *L*.

The accuracy goal is then changed: can they get within one vertical meter? Yes. If they are anywhere within seven horizontal meters of *p*, their altitude will always remain within one meter from the target *L*. In summary, to say that the traveler's altitude approaches *L* as their horizontal position approaches *p*, is to say that for every target accuracy goal, however small it may be, there is some neighbourhood of *p* whose altitude fulfills 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

is to say that *ƒ*(*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.

Suppose *f* : **R** → **R** is defined on the real line and *p, L* ∈ **R**. One would say that **the limit of f, as x approaches p, is L** and written

or alternatively as:

- as (reads " tends to as tends to ")

if the following property holds:

- For every real
*ε*> 0, there exists a real*δ*> 0 such that for all real x, 0 < |*x*−*p*| <*δ*implies that |*f*(*x*) −*L*| <*ε*.^{[6]}

A more general definition applies for functions defined on subsets of the real line. Let (*a*, *b*) be an open interval in **R**, and *p* a point of (*a*, *b*). Let *f* be a real-valued function defined on all of (*a*, *b*)—except possibly at *p* itself. 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 0 < | *x* − *p* | < *δ* and *x* ∈ (*a*, *b*) implies that | *f*(*x*) − *L* | < *ε*.

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.

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,^{[2]} though in his definition of continuity, he used an infinitesimal 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.

Main article: One-sided limit |

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

or

respectively. If these limits exist at p and are equal there, then this can be referred to as ** the limit of f(x) at p**.

A formal definition is as follows. The limit of *f*(*x*) as *x* approaches *p* from above is *L* if, for every *ε* > 0, there exists a *δ* > 0 such that |*f*(*x*) − *L*| < *ε* whenever 0 < *x* − *p* < *δ*. The limit of *f*(*x*) as *x* approaches *p* from below is *L* if, for every *ε* > 0, there exists a *δ* > 0 such that |*f*(*x*) − *L*| < *ε* whenever 0 < *p* − *x* < *δ*.

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

Apart from open intervals, limits can be defined for functions on arbitrary subsets of **R**, as follows (Bartle & Sherbert 2000) : let *f* be a real-valued function defined on a subset *S* of the real line. Let *p* be a limit point of *S*—that is, *p* is the limit of some sequence of elements of *S* distinct from p. The limit of *f*, as *x* approaches *p* from values in *S*, is *L,* if for every *ε* > *0*, there exists a *δ* > *0* such that 0 < |*x* − *p*| < *δ* and *x* ∈ *S* implies that |*f*(*x*) − *L*| < *ε*.

This limit is often written as:

The condition that *f* be defined on *S* is that *S* be a subset of the domain of *f*. This generalization includes as special cases limits on an interval, as well as left-handed limits of real-valued functions (e.g., by taking *S* to be an open interval of the form ), and right-handed limits (e.g., by taking *S* to be an open interval of the form ). 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* can have limit 0 as x approaches 0 from above.

The definition of limit given here does not depend on how (or whether) *f* is defined at *p*. Bartle (1967) 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*:

- A number
*L*is the non-deleted limit of*f*as*x*approaches*p*if, for every*ε*>*0*, there exists a*δ*>*0*such that |*x*−*p*| <*δ*and*x*∈*Dm*(*f*) implies |*f*(*x*) −*L*| <*ε*.

The definition is the same, except that the neighborhood | *x* − *p* | < *δ* now includes the point *p*, in contrast to the deleted neighborhood 0 < | *x* − *p* | < *δ*. This makes the definition of a non-deleted limit less general. One of the advantages of working with non-deleted limits is that they allow to state the theorem about limits of compositions without any constraints on the functions (other than the existence of their non-deleted limits) (Hubbard (2015)).

Bartle (1967) notes that although by "limit" some authors do mean this non-deleted limit, deleted limits are the most popular. For example, Apostol (1974), Courant (1924), Hardy (1921), Rudin (1964), Whittaker & Watson (1902) all take "limit" to mean the deleted limit.

The function

has no limit at (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), but has a limit at every other *x*-coordinate.

The function

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

The function

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).

The functions

and

both have a limit at *x* = 0 and it equals 0.

The function

has a limit at any *x*-coordinate of the form , where *n* is any integer.

Suppose *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

if the following property holds:

- For every ε > 0, there exists a δ > 0 such that d
_{B}(*f*(*x*),*L*) < ε whenever 0 <*d*_{A}(*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*, and even if *f*(*p*) is defined it need not be equal to *L*.

An alternative definition using the concept of neighbourhood is as follows:

if, for every neighbourhood *V* of *L* in *B*, there exists a neighbourhood *U* of *p* in *A* such that *f*(U ∩ M − {*p*}) ⊆ *V*.

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

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* ".

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. 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 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.

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*.

Let , and .

**The limit of f as x approaches infinity is L**, denoted

means that for all , there exists *c* such that
whenever *x* > *c*. Or, symbolically:

- .

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

means that for all there exists *c* such that whenever *x* < *c*. Or, symbolically:

- .

For example,

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 , and . The statement **the limit of f as x approaches a is infinity**, denoted

means that for all there exists such that whenever

These ideas can be combined in a natural way to produce definitions for different combinations, such as

For example,

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**, - a neighborhood of ∞ is defined to contain an interval (
*c*, ∞] where*c*∈**R**, and - 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.

Many authors^{[8]} 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** ∪ {−∞, +∞} and the projectively extended real line is **R** ∪ {∞} where a neighborhood of ∞ is a set of the form {*x*: |*x*| > *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, does not possess a central limit (which is normal):

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

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 , namely, it is convenient for to be considered true. Such zeroes can be seen as an approximation to infinitesimals.

There are three basic rules for evaluating limits at infinity for a rational function *f*(*x*) = *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.

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* : **R**^{2} → **R**,

if

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

where ||(*x*,*y*) − (*p*,*q*)|| represents the Euclidean distance. This can be extended to any number of variables.

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 (*x*_{n}) in*X*− {*p*} that converges to*p*, the sequence*f*(*x*_{n}) 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*.

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:

if, and only if, for all sequences (with not equal to *a* for all *n*) converging to the sequence converges to . 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 to converge to 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 *p*
if for every sequence ∈ *Dm*(*f*) \ {*a*} (so that for all *n*, is not equal to *a*) that converges to *a*, the sequence converges to . This is the same as the definition of a sequential limit in the preceding section obtained by regarding the subset *Dm*(*f*) of **R** as a metric space with the induced metric.

In non-standard calculus the limit of a function is defined by:

if and only if for all , is infinitesimal whenever is infinitesimal. Here are the hyperreal numbers and 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.^{[9]} 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.^{[10]}
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".^{[11]}

At the 1908 international congress of mathematics F. Riesz introduced an alternate way defining limits and continuity in concept called "nearness". A point is defined to be near a set if for every there is a point so that . In this setting the

if and only if for all , is near whenever is near . Here is the set . This definition can also be extended to metric and topological spaces.

The notion of the limit of a function is very closely related to the concept of continuity. A function *ƒ* 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*:

(We have here assumed that *c* is a limit point of the domain of *f*.)

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* : *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., *,* *,* *,* *,* ) 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).

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*> 0*q*× ∞ = −∞ if*q*< 0*q*/ ∞ = 0 if*q*≠ ∞ and*q*≠ −∞- ∞
^{q}= 0 if*q*< 0 - ∞
^{q}= ∞ if*q*> 0 *q*^{∞}= 0 if 0 <*q*< 1*q*^{∞}= ∞ if*q*> 1*q*^{−∞}= ∞ if 0 <*q*< 1*q*^{−∞}= 0 if*q*> 1

(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:

- 0 / 0
- ±∞ / ±∞
- 0 × ±∞
- ∞ + −∞
- 0
^{0} - ∞
^{0} - 1
^{±∞}

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

In general, from knowing that

- and ,

it does *not* follow that . 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 such that if then ).

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

Since the value at *f*(0) is a removable discontinuity,

- for all .

Thus, the naïve chain rule would suggest that the limit of *f*(*f*(*x*)) is 0. However, it is the case that

and so

- for all .

Main article: List of limits |

For a nonnegative integer and constants and ,

This can be proven by dividing both the numerator and denominator by . 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.

Main article: 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:

- or , and
- and are differentiable over , and
- for all , and
- exists,

then:

Normally, the first condition is the most important one.

For example:

Specifying an infinite bound on a summation or integral is a common shorthand for specifying a limit.

A short way to write the limit is . An important example of limits of sums such as these are series.

A short way to write the limit is .

A short way to write the limit is .