An initial definition provides an intuitive understanding of limits.
Definition. Suppose is defined when is near the number (This means that is defined around , except possibly at itself.). Then we write
and say “the limit of , as approaches , equals ” if we can make the values of arbitrarily close to by restricting to be sufficiently close to but not equal to .
When we consider both the left and right limits, we define a limit to exist if and only if both the left and right limits exist and satisfy the condition:
The the precise definition of limits uses the language of and .
Definition. exists if for any , there exists such that $ |
f(x)-L |
<\varepsilon |
x-a |
<\delta |
x-a |
<\delta\Rightarrow |
f(x)-L |
<\varepsilon$). |