When I started learning Multivariable calc first we went back and developed a new notion of derivative as a linear approximation. And what we came up with was F(a+h)=F(a)+mh+E(h) * where m is the derivative. Basically the function minus the line tangent to point a. However there is a requirement that E(h)/h->0 as h->0. In the book and my proff say that we want E(h) to go faster to 0 then h, but that's not very mathematical! I asked my proff and he said that we don't have to devide by h it just makes approx better. The reason I am heaving trouble is because if I were to discovering the derivative for the 1st time using line approx (without knowing limit way) the condition for E(h)/h->0 as h->0 would not be obvious. It seems out of no wear. Why cant E(h)->0 as h->0 on its own without heaving to worry what happens when to E(h)/h as h->0. Could some guide me through the logic for the requirement of E(h)/h->0 as h->0 Thank you. PS: I know that if we rearrange * then E(h)/h->0 as h->0 works out but im asking if we just focus in this linear approx way only!