# (ε, δ)-definition of limit

(ε, δ)-definition of limit

In calculus, the 19th-century German mathematician Karl Weierstrass formulated the (ε, δ)-definition of limit ("epsilon-delta definition of limit").

Informal statement

Let "&fnof;" be a function. To say that

: $lim_\left\{x o c\right\}f\left(x\right) = L ,$

means that "&fnof;"("x") can be made as close as desired to "L" by making "x" close enough, but not equal, to "c".

How close is "close enough to "c" depends on how close one wants to make "&fnof;"("x") to "L". It also of course depends on which function "&fnof;" is and on which number "c" is. The positive number "&epsilon;" (epsilon) is how close one wants to make "&fnof;"("x") to "L"; one wants the distance to be no more than "&epsilon;". The positive number "&delta;" is how close one will make "x" to "c"; if the distance from "x" to "c" is less than "&delta;" (but not zero), then the distance from "&fnof;"("x") to "L" will be less than "&epsilon;". Thus "&epsilon;" depends on "&delta;". The limit statement means that no matter how small "&epsilon;" is made, "&delta;" can be made small enough.

Precise statement

The (ε, δ)-definition of the limit of a function is as follows:

Let "&fnof;" be a function defined on an open interval containing "c" (except possibly at "c") and let "L" be a real number. Then the formula

: $lim_\left\{x o c\right\}f\left(x\right) = L ,$

means

:for each real "&epsilon;" > 0 there exists a real "&delta;" > 0 such that for all "x" with 0 < |"x" − "c"| < "&delta;", we have |"&fnof;"("x") − "L"| < "&epsilon;".

A function "&fnof;" is said to be continuous at "c" if

: $lim_\left\{x o c\right\} f\left(x\right) = f\left(c\right).$

If the condition 0 < |"x" − "c"| is left out of the definition of limit, then requiring "&fnof;"("x") to have a limit at "c" would be the same as requiring "&fnof;"("x") to be continuous at "c".

Similarly, a function "&fnof;" is said to be uniformly continuous on an interval "I" if

:for each real "&epsilon;" > 0 there exists a real "&delta;" > 0 such that for all real numbers "x" and "y" in "I" with |"x" − "y"| < "&delta;", we have |"&fnof;"("x") − "&fnof;"("y")| < "&epsilon;".

The difference between uniform continuity on an interval, and continuity at all points in the interval separately, is that with uniform continuity, the "&delta;" that is small enough may be taken to be the same at all points in the interval. As an example sin(1/"x") is a continuous function of "x" at every point in the interval (0, &infin;), but it is not uniformly continuous on that interval.

In the study of non-standard calculus, it is possible to state the definition of the limit of a function on the hyperreals without using quantifiers. Alternative definitions without quantifiers may be found at non-standard calculus.

Limit of sequence

For a sequence of real numbers $\left\{x_n|nin mathbb\left\{N\right\}\right\};$,

A real number "L" is called the limit of the sequence, written symbolically as

::$lim_\left\{n o infty\right\} x_n =L,$

if and only if for every real number "&epsilon;" > 0, there exists a natural number "N", such that for every "n" &gt; "N" we have |"x""n" − "L"| &lt; &epsilon;

An alternative quantifier-free definition appears at non-standard calculus.

ee also

*list of calculus topics

Wikimedia Foundation. 2010.

### См. также в других словарях:

• Limit state design — (LSD) refers to a design methodology used in structural engineering. The methodology is in fact a modernization and rationalization of engineering knowledge which was well established prior to the adoption of LSD. Beyond the concept of a limit… …   Wikipedia

• limit# — limit n Limit, bound, confine, end, term are comparable when they mean an actual or imaginary line beyond which a thing does not or cannot extend. Limit is the most inclusive of these terms because it carries no necessary implication of number,… …   New Dictionary of Synonyms

• Limit — Lim it (l[i^]m [i^]t), v. t. [imp. & p. p. {Limited}; p. pr. & vb. n. {Limiting}.] [F. limiter, L. limitare, fr. limes, limitis, limit; prob. akin to limen threshold, E. eliminate; cf. L. limus sidelong.] To apply a limit to, or set a limit for;… …   The Collaborative International Dictionary of English

• Limit (Jugendzeitschrift) — Limit war eine Jugendzeitschrift der Walt Disney Company. Herausgegeben wurde sie von 1992 bis 1998 durch den Ehapa Verlag in Deutschland und durch den Egmont Verlag in Österreich. Sie erschien monatlich. Das Magazin sollte Jungen erreichen, die… …   Deutsch Wikipedia

• Limit — Lim it (l[i^]m [i^]t), n. [From L. limes, limitis: cf. F. limite; or from E. limit, v. See {Limit}, v. t.] 1. That which terminates, circumscribes, restrains, or confines; the bound, border, or edge; the utmost extent; as, the limit of a walk, of …   The Collaborative International Dictionary of English

• Limit loads — Limit load is the maximum load that a structure can safely carry. It s the load at which the structure is in a state of incipient plastic collapse. As the load on the structure increases, the displacements increases linearly in the elastic range… …   Wikipedia

• Limit — steht für eine Mengengrenze oder Betragsgrenze, siehe Grenzwert einen Begriff aus dem Pokerspiel, siehe Liste von Pokerbegriffen einen Orderzusatz einer Wertpapierorder in Form einer Kursober oder untergrenze, siehe Limitorder Limit… …   Deutsch Wikipedia

• limit — [lim′it] n. [OFr limite < L limes (gen. limitis), border, frontier] 1. the point, line, or edge where something ends or must end; boundary or border beyond which something ceases to be or to be possible 2. [pl.] bounds; boundary lines 3. the… …   English World dictionary

• Limit (Roman) — Limit ist ein Roman des deutschen Schriftstellers Frank Schätzing, der Anfang Oktober 2009 im Verlag Kiepenheuer Witsch erschien. Zentrales Thema des Thrillers ist der Abbau des Heliumisotops Helium 3 auf dem Mond, welches zur Energiegewinnung… …   Deutsch Wikipedia

• limit — I noun ambit, border, bound, boundary, boundary line, circumscriptio, circumscription, extreme boundary final point, finis, fringe, frontier, furthest point, line of demarcation, outer edge, outer line, outer point, perimeter, rim, terminus,… …   Law dictionary

• Limit — Sn Grenze, Preisrahmen erw. fach. (20. Jh.) Entlehnung. Entlehnt aus ne. limit, dieses aus frz. limite f., aus l. līmes (limitis) m. Grenzlinie, Querweg, Rain . Schon früher aus dem Französischen entlehnt ist die verbale Ableitung limitieren.… …   Etymologisches Wörterbuch der deutschen sprache

### Поделиться ссылкой на выделенное

##### Прямая ссылка:
Нажмите правой клавишей мыши и выберите «Копировать ссылку»