limit

Dictionary: lim·it (lĭm'ĭt) pronunciation

The point, edge, or line beyond which something cannot or may not proceed.
limits The boundary surrounding a specific area; bounds: within the city limits.
A confining or restricting object, agent, or influence.
The greatest or least amount, number, or extent allowed or possible: a withdrawal limit of $200; no minimum age limit.
Games. The largest amount which may be bet at one time in games of chance.
(Abbr. lim) Mathematics. A number or point L that is approached by a function f(x) as x approaches a if, for every positive number ε, there exists a number δ such that |f(x)−L| < ε if 0 < |x−a| < δ. Also called limit point, point of accumulation.
Informal. One that approaches or exceeds certain limits, as of credibility, forbearance, or acceptability: He is the limit of irresponsibility.

tr.v., -it·ed, -it·ing, -its.

To confine or restrict within a boundary or bounds.
To fix definitely; to specify.

[Middle English limite, from Old French, border, from Latin līmes, līmit-, border, limit.]

limitable lim'it·a·ble adj.

SYNONYMS limit, restrict, confine, circumscribe. These verbs mean to establish or keep within specified bounds. Limit refers principally to the establishment of a maximum beyond which a person or thing cannot or may not go: The Constitution limits the President's term of office to four years. To restrict is to keep within prescribed limits, as of choice or action: The sale of alcoholic beverages is restricted to those over 21. Confine suggests imprisonment, restraint, or impediment: The children were confined to the nursery. Circumscribe connotes an encircling or surrounding line that confines, especially narrowly: “A man . . . should not circumscribe his activity by any inflexible fence of rigid rules” (John Stuart Blackie).

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Banking Dictionary: Limit

1. Legal Lending Limit on loans to a single borrower. National banks and savings and loan associations cannot make unsecured loans greater than 15% of capital, and secured loans above 25% of capital.

2. Bank's own internal credit limit in loans to a single borrower, for example, a guidance line of credit. The guidance line is never disclosed.

3. Consumer's Credit Limit as in credit cards.

4. In Foreign Exchange a daily trading limit: the maximum amount a dealer is willing to trade or deposit with another bank. Also, a central bank's limit on long or short open positions.

5. Country Limit.

6. In electronic funds transfers, a Bilateral Credit Limit negotiated by two banks to prevent overdrawing a Reserve Account at a Federal Reserve Bank.

7. Limit up/limit down: in commodities markets and financial futures, the largest daily price change allowed by a futures exchange on Futures Contracts.

Thesaurus: limit

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noun

A demarcation point or boundary beyond which something does not extend or occur: bound² (often used in plural), confine (used in plural), end. See edge/center.
Either of the two points at the ends of a spectrum or range: extreme. See edge/center.
The boundary surrounding a certain area. bound² (used in plural), confine (used in plural), precinct (often used in plural). See limited/unlimited.
Something that limits or restricts: check, circumscription, constraint, cramp², curb, inhibition, limitation, restraint, restriction, stricture, trammel. See limited/unlimited.
The greatest amount or number allowed: ceiling, limitation, maximum. See limited/unlimited.
The ultimate point to which an action, thought, discussion, or policy is carried: end, extreme, length. See limited/unlimited.

verb

To place a limit on: circumscribe, confine, restrict. See limited/unlimited.
To fix the limits of: bound², delimit, delimitate, demarcate, determine, mark (off or out), measure. See limited/unlimited.

Antonyms: limit

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Definition: greatest extent
Antonyms: infinity, limitlessness, minimum

n

Definition: physical boundary
Antonyms: center, limitlessness

v

Definition: confine, restrict
Antonyms: allow, free, increase, let go, release, unbound

Dental Dictionary: limit

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Restriction.

Britannica Concise Encyclopedia: limit

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Mathematical concept based on the idea of closeness, used mainly in studying the behaviour of functions close to values at which they are undefined. For example, the function 1/x is not defined at x = 0. For positive values of x, as x is chosen closer and closer to 0, the value of 1/x begins to grow rapidly, approaching infinity as a limit. This interplay of action and reaction as the independent variable moves closer to a given value is the essence of the idea of a limit. Limits provide the means of defining the derivative and integral of a function.

For more information on limit, visit Britannica.com.

Columbia Encyclopedia: limit

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limit, in mathematics, value approached by a sequence or a function as the index or independent variable approaches some value, possibly infinity. For example, the terms of the sequence ¹/₂, ¹/₄, ¹/₈, ¹/₁₆,...are obviously getting smaller and smaller; since, if enough terms are taken, one can make the last term as small, i.e., as close to zero, as one pleases, the limit of this sequence is said to be zero. Similarly, the sequence 3, 5, 3¹/₂, 4¹/₂, 3³/₄, 4¹/₄, 3⁷/₈, 4¹/₈,...is seen to approach 4 as a limit. However, the sequences 1, 2, 4, 8, 16,...and 1, 2, 1, 2, 1, 2,...do not have limits. Frequently a sequence is denoted by giving an expression for the nth term, s_n; e.g., the first example is denoted by s_n=¹/₂ⁿ. The limit, s, of a sequence can then be expressed as lim s_n=s, or in the case of the example, lim ¹/₂ⁿ=0 (read “the limit of ¹/₂ⁿ as n approaches infinity is zero”). A sequence is a special case of a function. In many functions commonly encountered, the values of the independent variable (the domain) and those of the dependent variable (the range) may be any numbers, while for a sequence the domain is restricted to the positive integers, 1, 2, 3,.... The function y=¹/₂^x resembles the sequence used as an example, but note that x can take on values other than 1, 2, 3,...; thus we find not only lim ¹/₂^x=0 but also lim ¹/₂^x=4. A more precise definition of the limit of a function is: The function y=f(x) approaches a limit L as x approaches some number a if, for any positive number ε, there is a positive number δ such that |f(x)−L|<ε if 0<|x−a|<δ. Similarly, f(x) has the limit L as x becomes infinite if for any positive ε there is a δ such that |f(x)−L|<ε if |x|>δ.

Word Tutor: limit

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IN BRIEF: n. -The boundary of a specific area; As far as something can go.

Anyone who thinks the sky is the limit, has limited imagination. — Unknown

Wikipedia: Limit (mathematics)

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This article does not cite any references or sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (November 2008)

In mathematics, the concept of a "limit" is used to describe the behavior of a function as its argument or input either "gets close" to some point, or as the argument 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.

In formulas, limit is usually abbreviated as lim (see below).

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.

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
7 External links

Limit of a function

Main article: Limit of a function

Suppose ƒ(x) is a real-valued function and c is a real number. 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, 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)
1.947	1.995	1.999	$\Rightarrow \,\!$ undef $\Leftarrow$	2.0005	2.005	2.049

Thus, f(x) can be made arbitrarily close to the limit of 2 just by making x sufficiently close to 1.

Formal definition

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

Karl Weierstrass formally defined a limit as follows:

Let f be a real-valued function defined on an open interval of real numbers containing c (except possibly at c) and let L be a real number. Then

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

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,

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

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.

This formal definition of the limit of a function is sometimes called the epsilon-delta form because it uses the Greek letters delta (δ) and epsilon (ε). 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. An alternative definition without quantifiers can be found at non-standard calculus.

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

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,

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

For all x > S, f(x) is within ε of L

Formally, we have the definition

$\lim_{x \to \infty} f(x) = L$ if and only if for each ε > 0 there exists an S such that $|f(x) - L| < \varepsilon \text{ whenever } x > S.$

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

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.

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

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

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).

Useful identities

The following rules are valid if the limits on the right hand side exist (and are finite).

$\lim_{n \to c} S \sdot f(n) = S \sdot \lim_{n \to c} f(n)$ , where S is a scalar multiplier.
$\lim_{n \to c} b^{f(n)} =\displaystyle b^{\lim_{n \to c} f(n)}$ , where b is a positive real number.
$\lim_{n \to c} ( f(n) + g(n) ) = \lim_{n \to c} f(n) + \lim_{n \to c} g(n)$
$\lim_{n \to c} ( f(n) - g(n) ) = \lim_{n \to c} f(n) - \lim_{n \to c} g(n)$
$\lim_{n \to c} ( f(n) \sdot g(n) ) = \lim_{n \to c} f(n) \sdot \lim_{n \to c} g(n)$
$\lim_{n \to c} \frac{f(n)}{g(n)} = \frac{\lim_{n \to c} f(n)}{\lim_{n \to c} g(n)}$ , if the right hand side makes sense; i.e., the two limits on the right exist, and $\lim_{n \to c} g(n) \ne 0$ .

These rules may fail if any of the limits on the right hand side are undefined or infinite.

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

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

The first limit can be proven with the squeeze theorem. For 0 < x < π/2:

sin x < x < tan x .

Dividing everything by sin(x) yields

$1 < \frac{x}{\sin x} < \frac{\tan x}{\sin x}$

$1 < \frac{x}{\sin x} < \frac{1}{\cos x}$

$\lim_{x \to 0} \frac{1}{\cos x} = \frac{1}{1} = 1$

$\lim_{x \to 0} \frac{x}{\sin x} = 1$

$\lim_{x \to 0} \frac{\sin x}{x} = 1$

L'Hôpital's rule

This rule uses derivatives and has a conditional usage. (It can only be directly used on limits that "equal" 0/0 or ±∞/±∞. Other indeterminate forms require some algebraic manipulation usually involving setting the limit equal to y, taking the natural logarithm of both sides, and then using l'Hôpital's rule.)

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

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 and integrals

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

A short way to write the limit $\lim_{n \to \infty} \int_{a}^{n} f(x) \; dx$ is $\int_{a}^{\infty} f(x) \; dx$ .

A short way to write the limit $\lim_{n \to -\infty} \int_{n}^{b} f(x) \; dx$ is $\int_{-\infty}^{b} f(x) \; dx$ .

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.

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

Limit in category theory

Main article: Limit (category theory)

External links

Mathwords: Limit

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

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Translations: Limit

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Dansk (Danish)
n. - grænse, maksimum, minimum, limitum, tolerance, grænseværdi
v. tr. - begrænse, indskrænke, limitere

idioms:

over the limit over grænsen, over stregen
within limits inden for visse grænser

Nederlands (Dutch)
limiet, grens, beperking, quota, paal en perk, (mv) grenzen, toppunt, het uiterste, begrenzen, beperken, limiteren

Français (French)
n. - limite, limitation
v. tr. - limiter, se limiter à (faire)

idioms:

be the limit dépasser les bornes
over the limit au-delà de la limite
within limits dans la limite de

Deutsch (German)
n. - Grenzwert, Grenze, Beschränkung
v. - begrenzen, einschränken

idioms:

be the limit [einfach] unmöglich sein
over the limit über dem Grenzwert
within limits innerhalb gewisser Grenzen

Ελληνική (Greek)
n. - όριο, σύνορο, πέρας
v. - περιορίζω/-ομαι, θέτω όρια

idioms:

over the limit πάνω από το όριο
within limits μέσα σε (λογικά) όρια

Italiano (Italian)
limitare, limite

idioms:

off limits vietato
over the limit oltre i limiti
within limits con moderazione

Português (Portuguese)
n. - limite (m)
v. - limitar

idioms:

off limits fora dos limites
over the limit acima do limite
within limits com moderação

Русский (Russian)
ограничивать, служить границей, предел

idioms:

off limits въезд запрещен, участие запрещено, вне границ
over the limit за пределами
within limits в пределах чего-л.

Español (Spanish)
n. - límite
v. tr. - limitar

idioms:

be the limit ser intolerable
over the limit exceder el límite
within limits dentro de ciertos límites

Svenska (Swedish)
n. - gräns, yttersta gräns, (mat.) gränsvärde, limit
v. - begränsa, sätta (en) gräns för, inskränka, (hand.) limitera

中文（简体）(Chinese (Simplified))
界限, 限制, 限度, 限定

idioms:

over the limit 超过限度
within limits 适当地

中文（繁體）(Chinese (Traditional))
n. - 界限, 限制, 限度
v. tr. - 限制, 限定

idioms:

over the limit 超過限度
within limits 適當地

한국어 (Korean)
n. - 극한점, 한계, 경계선
v. tr. - ~에 한계를 설정하다, 한정하다, 제한하다

idioms:

over the limit 한계를 넘어

日本語 (Japanese)
n. - 限界, 制限, 境界, 我慢の極限
v. - 制限する

idioms:

off limits 立ち入り禁止区域
over the limit 限度外の
within limits 適度に

العربيه (Arabic)
‏(الاسم) حد, نهايه, حد أقصى (فعل) حدد, قيد, حصر‏

עברית (Hebrew)
n. - ‮גבול, תחום, מגבלה‬
v. tr. - ‮צימצם, הגביל‬

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	You continually charge things to your credit card with 1000 limit and spend more than limit in a given cycle and to avoid going over limit you keep making early payments is this bad for credit score? Read answer...
	Will your credit score improve if you lower the credit limit on a credit card Current limit is at 12000 with a balance of 4900. What if the limit is lowered to 5000? Read answer...
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Help us answer these

	How does a weight limit translate into a momentum limit in other words if a weight limit is 100 lbs you weigh 100 lbs but you jump up and down would that break the support structure?
	There are different situations when a limit does not exist For example a sequence could have no limit or a sequence could have a limit to infinity Whats the differences between the two situations?
	The elevator has a limit of 16 ppl - There is a weight limit of 2500 lbs - Assume that the avg weight is 150 lbs SD is 27 - what is probability that a random sample of 16 ppl will exceed weight limit?

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		Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2007. Published by Houghton Mifflin Company. All rights reserved. Read more
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		Thesaurus. Roget's II: The New Thesaurus, Third Edition by the Editors of the American Heritage® Dictionary Copyright © 1995 by Houghton Mifflin Company. Published by Houghton Mifflin Company. All rights reserved. Read more
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		Dental Dictionary. Mosby's Dental Dictionary. Copyright © 2004 by Elsevier, Inc. All rights reserved. Read more
		Britannica Concise Encyclopedia. Britannica Concise Encyclopedia. © 2006 Encyclopædia Britannica, Inc. All rights reserved. Read more
		Columbia Encyclopedia. The Columbia Electronic Encyclopedia, Sixth Edition Copyright © 2003, Columbia University Press. Licensed from Columbia University Press. All rights reserved. www.cc.columbia.edu/cu/cup/ Read more
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limit

Contents

Limit of a function

Formal definition

Limit of a function at infinity

Limit of a sequence

Useful identities

Limits of extra interest

L'Hôpital's rule

Summations and integrals

Topological net

Limit in category theory

See also

External links