Limit (mathematics)

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In mathematics, the concept of a "limit" is used to describe the behavior of a function as its argument either gets "close" to some point, or as it becomes arbitrarily large; or the behavior of a sequence's elements, as their index increases indefinitely. Limits are used in calculus and other branches of mathematical analysis to define derivatives and continuity.

The concept of the "limit of a function" is further generalized to the concept of topological net, while the limit of a sequence is closely related to limit and direct limit in category theory.

Mathematics students usually first encounter limits in introductory calculus classes, and understanding the detailed concept often presents a stumbling block. While this article does have some elementary exposition — along with information about how limits are treated in more advanced branches of mathematics — readers seeking an introductory explanation might look at the Wikibooks Calculus section about limits, as it contains a more general approach.

1 Limit of a function
- 1.1 Formal definition
- 1.2 Limit of a function at infinity
2 Limit of a sequence
3 Topological net
4 Limit in category theory
5 See also

[edit] Limit of a function

Main article: Limit of a function

Suppose f(x) is a real function and c is a real number. The expression:

$\lim_{x \to c}f(x) = L$

means that f(x) can be made to be as close to L as desired by making x sufficiently close to c. In that case, we say that "the limit of f of x, as x approaches c, is L". Note that this statement can be true even if $f(c) \neq L$ . Indeed, the function f(x) need not even be defined at c. Two examples help illustrate this.

Consider $f(x) = \frac{x}{x^2 + 1}$ as x approaches 2. In this case, f(x) is defined at 2 and equals its limit of 0.4:

f(1.9)	f(1.99)	f(1.999)	f(2)	f(2.001)	f(2.01)	f(2.1)
0.4121	0.4012	0.4001	$\Rightarrow$ 0.4 $\Leftarrow$	0.3998	0.3988	0.3882

As x approaches 2, f(x) approaches 0.4 and hence we have $\lim_{x\to 2}f(x)=0.4$ . In the case where $f(c) = \lim_{x\to c} f(x)$ , f is said to be continuous at x = c. But it is not always the case. Consider

$g(x)=\left\{\begin{matrix} \frac{x}{x^2+1}, & \mbox{if }x\ne 2 \\ \\ 0, & \mbox{if }x=2. \end{matrix}\right.$

The limit of g(x) as x approaches 2 is 0.4 (just as in f(x)), but $\lim_{x\to 2}g(x)\neq g(2)$ ; g is not continuous at x = 2.

Or, consider the case where f(x) is undefined at x = c.

$f(x) = \frac{x - 1}{\sqrt{x} - 1}$

In this case, as x approaches 1, f(x) is undefined at x = 1 but the limit equals 2:

f(0.9)	f(0.99)	f(0.999)	f(1.0)	f(1.001)	f(1.01)	f(1.1)
1.95	1.99	1.999	$\Rightarrow$ undef $\Leftarrow$	2.001	2.010	2.10

Thus, x can get as close to 1, so long as it is not equal to 1, so that the limit of $f (x)$ is 2.

[edit] Formal definition

A limit is formally defined as follows: Let $f\,$ be a function defined on an open interval containing $c\,$ (except possibly at $c\,$ ) and let $L\,$ be a real number.

$\lim_{x \to c}f(x) = L$

means that

for each real $\epsilon\ >0$ there exists a real $\delta\ >0$ such that for all $x\,$ where $0<|x-c|< \delta\$ , $| f (x)-L|< \epsilon\$ .

[edit] Limit of a function at infinity

A related concept to limits as x approaches some finite number is the limit as x approaches positive or negative infinity. This does not literally mean that the difference between x and infinity becomes small, since infinity is not a real number; rather, it means that x either grows without bound positively (positive infinity) or grows without bound negatively (negative infinity).

For example, consider $f(x) = \frac{2x}{x + 1}$ .

f(100) = 1.9802
f(1000) = 1.9980
f(10000) = 1.9998

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 just by picking x sufficiently large. In this case, we say that the limit of f(x) as x approaches infinity is 2. In mathematical notation, one writes

$\lim_{x \to \infty} f(x) = 2.$

Formally, we have the definition

$\lim_{x \to \infty} f(x) = c$ if and only if for each

ε > 0

there exists an

n

such that

| f (x) - c | < ε

whenever

x > n .

Note that the n in the definition will generally depend on $ε$ . A similar definition applies for $\lim_{x \to -\infty} f(x)=c$ .

If one considers the domain of $f\,$ to be the extended real number line, then the limit of a function at infinity can be considered as a special case of limit of a function at a point.

[edit] Limit of a sequence

Main article: Limit of a sequence

Consider the following sequence: 1.79, 1.799, 1.7999,... We could observe that the numbers are "approaching" 1.8, the limit of the sequence.

Formally, suppose x₁, x₂, ... is a sequence of real numbers. We say that the real number L is the limit of this sequence and we write

$\lim_{n \to \infty} x_n = L$

if and only if

for every real number

ε > 0

there exists a natural number

n 0

(which will depend on

ε

) such that for all

n > n 0

we have

| x n - L | < ε

Intuitively, this means that eventually all elements of the sequence get as close as we want to the limit, since the absolute value $| x n - L |$ can be interpreted as the "distance" between x_n and L. Not every sequence has a limit; if it does, we call it convergent, otherwise 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 of a sequence is simply the limit at infinity of a function defined on natural numbers. On the other hand, a limit of a function f at x, if it exists, is the same as the limit of the sequence $x n = f (x + 1 / n)$ .

[edit] Topological net

Main article: Net (topology)

All of the above notions of limit can be unified and generalized to arbitrary topological spaces by introducing topological nets and defining their limits. The article on nets elaborates on this.

An alternative is the concept of limit for filters on topological spaces.