We need to find a such that, for all

we have that,

The delta inequality is more conviently written as,

The second inequality can be written as,

Thus,

Here is the problem, I have no idea how to do this delta-epsilon. In fact, this limit is completely proved otherwise.

Below I have attach a beautiful hand drawn diagram that shows for the delta-epsilon statement is true.

The thin red line shows the curve

The thick blue and purple region show that it was squeezed between that area.