How to deduce the ODE from difference equation?

Question

Consider a difference equation $$ x(t+h)=x(t)+hf(t)+\frac{h^2}{2}g(t), $$ where $h$ is infinitesimal, $f,g$ are given functions of time. Let $h\to0$, we get $$ x'(t)=f(t). $$ But if I substitute Taylor expansion into L.H.S., I get $$ x'(t)=f(t),\quad x''(t)=g(t), $$ so $$ f'(t)=g(t). $$ But $f,g$ are arbitrary functions. What's wrong?