I have seen a limit formulation of steepest ascent (descent if we consider minimum):
$$\nabla f(x)\propto \lim_{\epsilon\rightarrow 0} \arg \max_{d:||d||=\epsilon} f(x+d)$$
So far I have: $f(x+d)\approx f(x)+d^T\nabla f(x)$ and $\arg\max_{d:||d||=\epsilon}f(x+d) = \frac{\nabla f(x)}{||\nabla{f(x}||}\epsilon$. Hmm. In the limit it equals 0?
I have seen formulations where $d$ is of unit length and where one assumes that the distance from f(x) is not far, but I do not get the above limit definition.