In mathematics, a limit is the value that a function (or sequence) "approaches" as the input (or index) "approaches" some value.^{[1]} Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.
The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory.
In formulas, a limit of a function is usually written as
and is read as "the limit of f of x as x approaches c equals L". The fact that a function f approaches the limit L as x approaches c is sometimes denoted by a right arrow (->), as in:
which reads " tends to as tends to ".^{[2]}
Suppose f is a real-valued function and c is a real number. Intuitively speaking, the expression
means that f(x) can be made to be as close to L as desired, by making x sufficiently close to c.^{[3]} In that case, the above equation can be read as "the limit of f of x, as x approaches c, is L".
Augustin-Louis Cauchy in 1821,^{[4]} followed by Karl Weierstrass, formalized the definition of the limit of a function which became known as the (?, ?)-definition of limit. The definition uses ? (the lowercase Greek letter epsilon)^{[2]} to represent any small positive number, so that "f(x) becomes arbitrarily close to L" means that f(x) eventually lies in the interval (L - ?, L + ?), which can also be written using the absolute value sign as || < ?.^{[4]} The phrase "as x approaches c" then indicates that we refer to values of x, whose distance from c is less than some positive number ? (the lower case Greek letter delta)--that is, values of x within either (c - ?, c) or (c, c + ?), which can be expressed with 0 < || < ?. The first inequality means that the distance between x and c is greater than 0 and that x ? c, while the second indicates that x is within distance ? of c.^{[4]}
The above definition of a limit is true even if f(c) ? L. Indeed, the function f need not even be defined at c.
For example, if
then f(1) is not defined (see indeterminate forms), yet as x moves arbitrarily close to 1, f(x) correspondingly approaches 2:^{[5]}
f(0.9) | f(0.99) | f(0.999) | f(1.0) | f(1.001) | f(1.01) | f(1.1) |
1.900 | 1.990 | 1.999 | undefined | 2.001 | 2.010 | 2.100 |
Thus, f(x) can be made arbitrarily close to the limit of 2--just by making x sufficiently close to 1.
In other words, .
This can also be calculated algebraically, as for all real numbers x ? 1.
Now, since x + 1 is continuous in x at 1, we can now plug in 1 for x, leading to the equation .
In addition to limits at finite values, functions can also have limits at infinity. For example, consider the function
where:
As x becomes extremely large, the value of f(x) approaches 2, and the value of f(x) can be made as close to 2 as one could wish--by making x sufficiently large. So in this case, the limit of f(x) as x approaches infinity is 2, or in mathematical notation,
Consider the following sequence: 1.79, 1.799, 1.7999,... It can be observed that the numbers are "approaching" 1.8, the limit of the sequence.
Formally, suppose a_{1}, a_{2}, ... is a sequence of real numbers. One can state that the real number L is the limit of this sequence, namely:
which is read as
if and only if
Intuitively, this means that eventually, all elements of the sequence get arbitrarily close to the limit, since the absolute value || is the distance between a_{n} and L. Not every sequence has a limit; if it does, then it is called convergent, and if it does not, then it is divergent. One can show that a convergent sequence has only one limit.
The limit of a sequence and the limit of a function are closely related. On one hand, the limit as n approaches infinity of a sequence {a_{n}} is simply the limit at infinity of a function a(n)--defined on the natural numbers {n}. On the other hand, if X is the domain of a function f(x) and if the limit as n approaches infinity of f(x_{n}) is L for every arbitrary sequence of points {x_{n}} in {X - {x_{0}}} which converges to x_{0}, then the limit of the function f(x) as x approaches x_{0} is L.^{[7]} One such sequence would be {x_{0} + 1/n}.
In non-standard analysis (which involves a hyperreal enlargement of the number system), the limit of a sequence can be expressed as the standard part of the value of the natural extension of the sequence at an infinite hypernatural index n=H. Thus,
Here, the standard part function "st" rounds off each finite hyperreal number to the nearest real number (the difference between them is infinitesimal). This formalizes the natural intuition that for "very large" values of the index, the terms in the sequence are "very close" to the limit value of the sequence. Conversely, the standard part of a hyperreal represented in the ultrapower construction by a Cauchy sequence , is simply the limit of that sequence:
In this sense, taking the limit and taking the standard part are equivalent procedures.
A formal definition of convergence can be stated as follows. Suppose as goes from to is a sequence that converges to , with for all . If positive constants and exist with
then as goes from to converges to of order , with asymptotic error constant .
Given a function with a fixed point , there is a nice checklist for checking the convergence of the sequence .
then there is linear convergence | |
series diverges | |
then there is at least linear convergence and maybe something better, the expression should be checked for quadratic convergence |
then there is quadratic convergence provided that is continuous | |
then there is something even better than quadratic convergence | |
does not exist | then there is convergence that is better than linear but still not quadratic |
^{[8]}
Limits can be difficult to compute. There exist limit expressions whose modulus of convergence is undecidable. In recursion theory, the limit lemma proves that it is possible to encode undecidable problems using limits.^{[9]}