Precise Definition of a Limit
Here is the precise definition for a limit:
Let be defined on an open interval containing except possibly at itself. We say
The limit of as approaches is the number
and we write
if
For every number , there is some number such that
.
The last mathematical expression above might be interpreted: Whenever is within of (but not actually equal to ), is within of .
In this figure, a random will be given, and you must find a corresponding . NOTE: you may have to zoom in the window to select a that is small enough. As you zoom in, the slider to pick your will adjust.