There are many integral representations of the Dirac delta. For example, it is well known that $$\delta_\epsilon(s):=\sqrt{\frac{1}{\epsilon\pi}}e^{-s^2/\epsilon}$$ acts like $\delta(0)$ for small $\epsilon$, i.e., for sufficiently nice test functions $f$ we have $$\lim_{\epsilon\to 0}\int_{\mathbb R}f(s)\delta_{\epsilon}(s)\,ds=f(0).$$
Is there a general recipe to prove asymptotics like this for more complicated functions? For example, numerical evidence suggests that $$g_\epsilon(s):=\frac{\log \left(\frac{\left(2+\sqrt{4-s^2}\right) \left(2-2\epsilon+\sqrt{(1-\epsilon ) \left(4-4\epsilon-s^2\right)}\right) \left(2-\epsilon-\sqrt{4-4\epsilon-s^2}\right) \left(2-\epsilon-\sqrt{\left(4-s^2\right) (1-\epsilon )}\right)}{\left(2-\sqrt{4-s^2}\right) \left(2-2\epsilon-\sqrt{(1-\epsilon ) \left(4-4\epsilon-s^2\right)}\right) \left(2-\epsilon+\sqrt{4-4\epsilon-s^2}\right) \left(2-\epsilon+\sqrt{\left(4-s^2\right) (1-\epsilon )}\right)}\right)}{4 \pi \epsilon }$$ acts like $\delta(0)$ when integrated over $[-2,2]$.
Contrary to the simple example above, I don't see the mechanism why this function favours small $s$ nor do I see why it approximately has unit integral.
So, are there general strategies to prove something like $$\lim_{\epsilon\to 0}\int_{\mathbb R}f(s)g_{\epsilon}(s)\,ds=f(0)?$$
My particular function has the particular feature that it is a difference quotient and I thought about exploiting this fact, but the limiting function does not seem to be integrable.