The limit of a function at is if for every , there exists a such that implies .
That is, if is within any arbitrary distance of whenever is sufficiently close, but not necessarily equal to, .
Note that represents the greatest distance between the values of the function and its limit, while represents the distance from the values of the function's inputs to .
The idea that the condition is not required allows for the possibility that the values of function approaches , but the actual value of need not be , or even defined.
If, by chance, we have , then we also say that is continuous at .
Sequential characterisation Edit
The limit of a function has an equivalent definition involving real sequences:
- Suppose , and is an arbitrary sequence such that
- Now suppose as
Ie, with and
So take , so by the above
So we have found a sequence
The first paragraph shows the standard definition implies the sequential characterisation, and the second paragraph (via proof of the contrapositive) shows that the sequential characterisation implies the standard definition. Therefore the two are equivalent and either may be used as the definition.