In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value.^[1] The concept of limit allows one to, in a complete space, define a new point from a Cauchy sequence of previously defined points.^[2] Limits are essential to calculus (and mathematical analysis in general) 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, limit is usually abbreviated as lim as in lim(a_n) = a or represented by the right arrow (→) as in a_n → a.

Contents [hide] 1 Limit of a function 2 Limit of a sequence 3 Convergence and fixed point 4 Topological net 5 Limit in category theory 6 See also 7 Notes 8 External links

[edit] Limit of a function

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

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, it can be stated that "the limit of f of x, as x approaches c, is L". Note that this statement can be true even if f(c) ≠ L. Indeed, the function f(x) need not even be defined at c.

For example, if

then f(1) is not defined, yet as x approaches 1, f(x) approaches 2:

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	⇒ undef ⇐	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.

Karl Weierstrass formalized the definition of the limit of a function into what became known as the (ε, δ)-definition of limit in the 19th century.

In addition to limits at finite values, functions can also have limits at infinity. For example, consider

• f(100) = 1.9900
• f(1000) = 1.9990
• f(10000) = 1.9999

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, the limit of f(x) as x approaches infinity is 2. In mathematical notation,

[edit] Limit of a sequence

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 x₁, x₂, ... is a sequence of real numbers. It can be stated that the real number L is the limit of this sequence, namely:

to mean

For every real number ε > 0, there exists a natural number n₀ such that for all n > n₀, |x_n − L| < ε.

Intuitively, this means that eventually all elements of the sequence get as close as needed to the limit, since the absolute value |x_n − L| is the distance between x_n and L. Not every sequence has a limit; if it does, it is called convergent, and if it does not, 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 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 f(a_n) where a_n is any arbitrary sequence whose limit is x, and where a_n is never equal to x. Note that one such sequence would be x + 1/n.

[edit] Convergence and fixed point

A formal definition of convergence can be stated as follows. Suppose p_n as n goes from 0 to is a sequence that converges to a fixed point p, with for all n. If positive constants λ and α exist with

then p_n as n goes from 0 to converges to p of order α, with asymptotic error constant λ

Given a function f(x) = x with a fixed point p, there is a nice checklist for checking the convergence of p.

1) First check that p is indeed a fixed point:

f(p) = p

2) Check for linear convergence. Start by finding . If....

	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

3) If it is found that there is something better than linear the expression should be checked for quadratic convergence. Start by finding If....

	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

^[3]

[edit] Topological net

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

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

[edit] Limit in category theory

[edit] See also

The Wikibook Calculus has a page on the topic of Limits

• Limit of a function
• Limit of a sequence
• One-sided limit
• Squeeze theorem
• List of limits

[edit] Notes

[edit] External links

• Weisstein, Eric W., "Limit" from MathWorld.
• Mathwords: Limit