for many years i have been doing derivatives, and its definition: lim f(x+h)-f(x) / h always seemed quite obvious to me, but then i started to think about it...
when we do derivatives through its definition, we usually arrive to something like (h(x+h))/h and we remove the h on the numerator and denominator, leaving x+h, when we replace h by 0 and get to x....but the question is, wont we be doing a violation when we place h by 0? If we condsidered the previous formula, replacing 0 by 0 would cause a 0/0 indetermination, which arises from the fact that the removal rule only applices if h!=0, so....what is that last result we get? The result of ignoring the fact that h cannot be 0 but at the same time assume it is 0?
Thanks in advance