# Equivalency between $\max \{ 0, 1 - \exp[g(x)]\}$ and $\max \{0, -g(x)\}$ (with err term)

\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...

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