The definition of limit: "lim x -> a f(x) = c " means for all epsilon > 0, there exists delta > 0 such that for all x, if 0 < |x-a|<delta then |f(x) - c| < epsilon.

The definition of derivative: f'(x) = lim h -> 0 (f(x+h) - f(x))/h

That is, for all epsilon > 0, there exists delta > 0 such that for all h, if 0 < |h| < delta then |(f(x+h) - f(x))/h - f'(x)| < epsilon.

At no point do we divide by 0. h never takes on the value 0.