we have that,
The delta inequality is more conviently written as,
The second inequality can be written as,
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.