Limit (mathematics) - Citizendia

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Behavior or behaviour (see spelling differences) refers to the actions or Reactions of an object or Organism, usually The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Logic, an argument is a Set of one or more Declarative sentences (or "propositions") known as the Premises along In Mathematics, a sequence is an ordered list of objects (or events The word index is used in variety of senses in Mathematics. In perhaps the most frequent sense an index is a Superscript Limits are used in calculus and other branches of mathematical analysis to define derivatives and continuity. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Analysis has its beginnings in the rigorous formulation of Calculus. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output

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. This article is about nets in Topological spaces and not about ε-nets in Metric spaces In Topology and related areas of Mathematics In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts In Mathematics, the direct limit (also called the inductive limit) is a general method of taking colimits of "directed families of objects" In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets

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

Limit of a function

Main article: Limit of a function

Definition: f(x)=lim (as n--> x) of f(n)

Suppose ƒ(x) is a real-valued function and c is a real number. In Mathematics, the limit of a function is a fundamental concept in Calculus and analysis concerning the behavior of that function near a particular In Mathematics, a function of a real variable is a Mathematical function whose domain is the Real line. In Mathematics, the real numbers may be described informally in several different ways The expression:

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

means that ƒ(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 ƒ of x, as x approaches c, is L". Note that this statement can be true even if $\scriptstyle f(c) \neq L$ . Indeed, the function ƒ(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, ƒ(x) approaches 0. 4 and hence we have $\scriptstyle \lim_{x\to 2}f(x)=0.4$ . In the case where $\scriptstyle f(c) = \lim_{x\to c} f(x)$ , ƒ is said to be continuous at x = c. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output 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 ƒ(x)), but $\scriptstyle \lim_{x\to 2}g(x)\neq g(2)$ ; g is not continuous at x = 2.

Or, consider the case where ƒ(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 (0/0) 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, f(x) can be made arbitrarily close to the limit of 2 just by making x sufficiently close enough to 1.

Formal definition

Karl Weierstrass formally defined a limit 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. Karl Theodor Wilhelm Weierstrass ( Weierstraß) ( October 31, 1815 &ndash February 19, 1897) was a German mathematician In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set In Mathematics, the real numbers may be described informally in several different ways

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

means that

for each real ε > 0 there exists a real δ > 0 such that for all x with 0 < |x − c| < δ, we have |f(x) − L| < ε.

or, symbolically,

$\forall \varepsilon > 0 \ \ \exists \delta > 0 \ \ \forall x (0 < |x - c| < \delta \ \rightarrow \ |f(x) - L| < \varepsilon).$

Compared to the informal discussion above, the fact that ε can be any arbitrarily small positive number corresponds to being able to bring f(x) as close to L as desired. The δ marks some "sufficiently close" distance for values of x from c such that f(x) stays within a distance less than ε from the limit L.

The formal definition of a limit is sometimes called the delta-epsilon form because it uses the Greek letters delta (δ) and epsilon (ε). The Greek alphabet (Ελληνικό αλφάβητο is a set of twenty-four letters that has been used to write the Greek language since the late 9th or early Delta (uppercase Δ, lowercase δ; Δέλτα Thelta is the fourth letter of the Greek alphabet. Epsilon (uppercase Ε, lowercase ε; Έψιλον is the fifth letter of the Greek alphabet, corresponding phonetically to a Close-mid front unrounded The use of the particular Greek letters δ and ε is merely traditional; the definition would, of course, be unchanged if different letters or symbols were used.

Caution: It should be noted that this definition provides a way to recognize a limit without providing a way to calculate it. One often needs to find a limit using informal methods, especially when f(x) is discontinuous at c, for example, when f is a ratio with a denominator that becomes 0 at c. One should check that the result actually meets the Weierstrass definition in such cases.

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. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness 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) = 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,

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

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

$|f(x) - c| < \varepsilon \text{ whenever } x > n.$

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

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. In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined In Mathematics, the affinely extended real number system is obtained from the Real number system R by adding two elements +∞ and &minus∞ (pronounced

Limit of a sequence

Main article: Limit of a sequence

Consider the following sequence: 1. The limit of a sequence is one of the oldest concepts in Mathematical analysis. 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. In Mathematics, a sequence is an ordered list of objects (or events In Mathematics, the real numbers may be described informally in several different ways 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. In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an

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. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. 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. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an 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).

Useful Identities

$\lim_{n \to c} S \sdot f(n) = S \sdot \lim_{n \to c} f(n)$ , where S is a scalar multiplier. In Mathematics, scalar multiplication is one of the basic operations defining a Vector space in Linear algebra (or more generally a module in
$\lim_{n \to c} b^{f(n)} = b^{\lim_{n \to c} f(n)}$ , where b is a constant.

The following rules are only valid if the limits on the righthand side exist and are finite.

$\lim_{n \to c} ( f(n) + g(n) ) = \lim_{n \to c} f(n) + \lim_{n \to c} g(n)$
$\lim_{n \to c} ( f(n) - g(n) ) = \lim_{n \to c} f(n) - \lim_{n \to c} g(n)$
$\lim_{n \to c} ( f(n) \sdot g(n) ) = \lim_{n \to c} f(n) \sdot \lim_{n \to c} g(n)$
$\lim_{n \to c} \frac{f(n)}{g(n)} = \frac{\lim_{n \to c} f(n)}{\lim_{n \to c} g(n)}$ , if the denominator containing the limit does not equal zero

If any of the limits in the righthand side is undefined or infinite, these rules do not necessarily work.

For example, $\lim_{n \to \infty} (3n+2) + (2-3n) = 4$ but $\lim_{n \to \infty} (3n+2) + \lim_{n \to \infty}(2-3n)$ is undefined.

Limits of extra interest

$\lim_{n \to 0} \frac{\sin n}{n} = 1$
$\lim_{n \to 0} \frac{1 - \cos n}{n} = 0$

l'Hôpital's rule

This rule uses derivatives and has a conditional usage. In Calculus, l'Hôpital's rule (also called Bernoulli 's rule; often misspelled l'Hospital) uses Derivatives to help compute limits It can only be used on indeterminate forms. In Calculus and other branches of Mathematical analysis, an indeterminate form is an Algebraic expression obtained in the context of Limits

$\lim_{n \to c} \frac{f(n)}{g(n)} = \lim_{n \to c} \frac{f'(n)}{g'(n)}$

For example: $\lim_{n \to 0} \frac{\sin (2n)}{\sin (3n)} = \lim_{n \to 0} \frac{2 \cos (2n)}{3 \cos (3n)} = \frac{2 \sdot 1}{3 \sdot 1} = \frac{2}{3}$

Summations

A short way to write the limit $\lim_{n \to \infty} \sum_{i=s}^{n} f(i)$ is $\sum_{i=s}^{\infty} f(i)$

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. This article is about nets in Topological spaces and not about ε-nets in Metric spaces In Topology and related areas of Mathematics Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. This article is about nets in Topological spaces and not about ε-nets in Metric spaces In Topology and related areas of Mathematics The article on nets elaborates on this.

An alternative is the concept of limit for filters on topological spaces. In Mathematics, a filter is a special Subset of a Partially ordered set.

Limit in category theory

Main article: Limit (category theory)

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