Limit (mathematics)

This is an overview of the idea of a limit in mathematics. For specific uses of a limit, see limit of a sequence and limit of a function.

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. Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives and integrals.

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.

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.

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

Limit of a function

Main article: Limit of a function

Whenever a point x is within δ units of p, f(x) is within ε units of L

For all x > S, f(x) is within ε of L

Suppose f(x) is a real-valued 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) ≠ L. Indeed, the function f(x) need not even be defined at c.

For example, if

$f(x) = \frac{x^2 - 1}{x - 1}$

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

$f(x) = {2x-1 \over x}$

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

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

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

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

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 we want 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, 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).

Convergence and fixed point

A formal definition of convergence can be stated as follows. Suppose ${ {p}_{n} }$ as $n$ goes from $0$ to $\infty$ is a sequence that converges to a fixed point $p$ , with ${p}_{n} \neq 0$ for all $n$ . If positive constants $\lambda$ and $\alpha$ exist with

$\lim_{n \rightarrow \infty } \frac{ \left | { p}_{n+1 } -p \right | }{ { \left | { p}_{n }-p \right | }^{ \alpha} } =\lambda$

then ${ {p}_{n} }$ as $n$ goes from $0$ to $\infty$ converges to $p$ of order $\alpha$ , with asymptotic error constant $\lambda$

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 $\left | f^\prime (p) \right |$ . If....

$\left \| f^\prime (p) \right \| \in (0,1]$	then we have linear convergence
$\left \| f^\prime (p) \right \| > 1$	series diverges
$\left \| f^\prime (p) \right \| =0$	then we have at least linear convergence and maybe something better, we should check for quadratic

3) If we find that we have something better than linear we should check for quadratic convergence. Start by finding $\left | f^{\prime\prime} (p) \right |$ If....

$\left \| f^{\prime\prime} (p) \right \| \neq 0$	then we have quadratic convergence provided that $f^{\prime\prime} (p)$ is continuous
$\left \| f^{\prime\prime} (p) \right \| = 0$	then we have something even better than quadratic convergence
$\left \| f^{\prime\prime} (p) \right \|$ does not exist	then we have convergence that is better than linear but still not quadratic

^[1]

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.

Limit in category theory

Notes

↑ Numerical Analysis, 8th Edition, Burden and Faires, Section 2.4 Error Analysis for Iterative Methods

External links

Mathwords: Limit