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)

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.