Given the following expression
$$ I(a) = \lim_{\Omega \to \mathbb{R}^3} \, \int_\Omega \nabla \left( \frac{1}{|x-a|} - \frac{1}{|x|} \right) dV $$
where $x$ and $a$ denote vectors in $\mathbb{R}^3$, I need to evaluate $I(a)$ – or understand where the evaluation fails.
First, you should note that evaluating the expression $I(a)$ fails for $x=0$ and $x=a$. This can be easily seen using the definition of the euclidean norm $|\cdot|$. Suppose $x=0$, then $|x|=\sqrt{0^2+0^2+0^2}=0$, and thus, $\frac{1}{|x|}=\frac{1}{0}$ wich is undefined. Something similar occurs for the norm of $|x-a|$ when $x=a$.
As for the actual value of the improper integral, you should note that for any vector $p\in\mathbb{R}^3$, you will have: $$\lim_{p\rightarrow 0}\frac{1}{|p|}=\infty$$ So, even if you integrate without the singularity vector points, the improper integral $I(a)$ might equate to an undetermined form $\infty-\infty$. Also there is the special case I(0)=0.