Does anyone knows how to show:
limit of f(x), x approaches a, equals L
if and only if
limit of f(x), x approaches a from the right = to limit of f(x), x approaches a from the left = L
Using the precise definition of limit.
Let be defined on an open interval containing except possibly at . With domain .
Let, .
That means,
.
We want to show,
.
Meaning,
.
.
Do you see? Note that can either be depending whether is larger than or not. But the point is that bpth of these quantieis are containted in the meaning of .