A Formal Definition of Limit:
Let be defined on a open interval about , except possibly at itself. We say that approaches the limit L as x approaches , and write
if for every number , there exists a corresponding number , such that for x,
Note: In the definition, is a function of , or in other words, depends on .
This definition is useful to prove important theorems about limits, that is, we can then easily evaluate the limits of complicated functions, whose limits are too cumbersome to evaluate from intuitive first principles.
Think of the definition as follows: We desire an output tolerance so that we need to find appropriate input given by
Let , and in the definition of limit. For any given , we have to find a number such that if . and x is within distance of
, that is, if
, then is within distance of , that is,
We find by working backwards from the -inequality.
, that is,
, that is,
Important note: The value of is not the only value that will make
imply . Any smaller positive will do as well. The definition does not ask for a “best” positive , just one that will work.
Any questions, comments are most welcome…