I've been thinking about proving the product rule, and I'm not sure if I can use limits in this way to do it:

A)

as hallsofivy points out, the last equality in the above line is wrong.. in physics or engineering such equalities are taken to be true since in most practical situations those are approximately true but strictly in many cases that will give huge error.
or should it be like this:

B)

As stated in the beginning of this post I was trying to prove the product rule, and consequently set this up:

A) feels a bit suspicious, but if it works(in the case above) it would make my task very easy, and B) seems a little pointless...

I'm feeling a bit puzzled at the moment, and any clarification would be appreciated.