Thursday, January 12, 2012

(ε, δ)-definition of limit

In calculus, the (ε, δ)-definition of limit ("epsilon-delta definition of limit") is a formalization of the notion of limit. It was given by Bernard Bolzano, in 1817, and in a less precise form by Augustin-Louis Cauchy.

Informal Statement

Let ƒ be a function. To say that
 \lim_{x \to c}f(x) = L \,
means that ƒ(x) can be made as close as desired to L by making the independent variable x close enough, but not equal, to the value 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 less 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 δ depends on ε. The limit statement means that no matter how small ε is made, δ can be made small enough.
The letters ε and δ can be understood as "error" and "distance", and in fact Cauchy used ε as an abbreviation for "error" in some of his work. In these terms, the error (ε) in the measurement of the value at the limit can be made as small as desired by reducing the distance (δ) to the limit point.
This definition also works for functions with more than one input value. In those cases, δ can be understood as the radius of a circle or sphere or higher-dimensional analogy, in the domain of the function and centered at the point where the existence of a limit is being proven, for which every point inside produces a function value less than ε from the value of the function at the limit point.

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 \to c}f(x) = L \,
for each real ε > 0 there exists a real δ > 0 such that for all x with 0 < |x − c| < δ, we have |ƒ(x) − L| < ε,
or, symbolically,
 \forall \varepsilon > 0\ \exists \ \delta > 0 : \forall x\ (0 < |x - c| < \delta \ \Rightarrow \ |f(x) - L| < \varepsilon).
The real inequalities exploited in the above definition were pioneered by Bolzano and Cauchy and formalized by Weierstrass.

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