I need to determine this limit value
$$ \lim_{r\rightarrow 0 } \frac{\partial}{\partial r }\left\{ r \sum_{\mathbb{k} } e^{i \vec{k} \cdot \vec{r} } [k^2 +1 - k (k^2 + 2 )^{1/2}] \right \} .$$
Essentially, I have a function $f(\vec{r})$ defined by the Fourier transform
$$ f(\vec{r}) = f(r) = \sum_{\mathbb{k} } e^{i \vec{k} \cdot \vec{r} } [k^2 +1 - k (k^2 + 2 )^{1/2}] . $$
The problem is to determine its behavior near the origin $\vec{r}=0 $. Here the sum is over a 3-dim lattice, i.e., $\vec{k}= (2\pi/L)(m,n,l)$ with $m,n,l$ being integers from $-\infty $ to $+\infty $.
It is expected that
$$f(r) = \frac{A}{r} + B + \text{higher oders of } r .$$
The problem is to determine $B$, which is a subleading term.