(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 "ƒ" be a function. To say that

: $lim\_\{x\; o\; c\}f(x)\; =\; L\; ,$

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

Precise statement

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

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

: $lim\_\{x\; o\; c\}f(x)\; =\; L\; ,$

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

: $lim\_\{x\; o\; c\}\; f(x)\; =\; f(c).$

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 $\{x\_n|nin\; mathbb\{N\}\};$,

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

::$lim\_\{n\; o\; infty\}\; x\_n\; =L,$

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