Outer Limits

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. more...

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Limits are used in calculus and other branches of mathematical analysis to define derivatives and continuity.

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.

Limit of a function

Suppose ƒ(x) is a real-valued function and c is a real number. The expression:

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 Consider f(x) = x/(x² + 1) as x approaches 2. In this case, f(x) is defined at 2 and equals its limit of 0.4:

As x approaches 2, ƒ(x) approaches 0.4 and hence we have The limit of g(x) as x approaches 2 is 0.4 (just as in ƒ(x)), but Or, consider the case where ƒ(x) is undefined at x = c.

In this case, as x approaches 1, f(x) is undefined at x = 1 but the limit equals 2:

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.

means that

Read more at Wikipedia.org