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$$\begin{align} &\max \{0, 1-\exp [\log f(x+h) - \log f(x)]\} \\ &= \max \{0, 1-\exp[\nabla \log f(x)h +o(h)] \} \\ &= \max\{0, -\nabla f(x) h\}+o(h) \end{align}$$

where $\forall x \in \mathbb{R}, f(x) > 0$, $f(x)$ is twice differentiable, and $h$ small.

I see the first step is by definition of differentials, but I have trouble seeing how the second equation is true...

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    $\begingroup$ The second step uses the series expansion of $\displaystyle e^x = \sum_{i = 0}^{\infty} \frac{x^k}{k!}$. $\endgroup$
    – sudeep5221
    Oct 18 '20 at 23:51
  • $\begingroup$ OH...........!!!! I will give it a try, thank you @sudeep5221 !!! $\endgroup$
    – user228809
    Oct 18 '20 at 23:52

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