For $x_0\in\mathbb{R}^n,n\geq 3$, let $\varphi(x)=\frac{1}{(n-2)\omega_n}\cdot\frac{1}{\|x-x_0\|^{n-2}}$, where $\omega_n$ is the surface area of the sphere $S_{n-1}=\{x\in\mathbb{R}^n;\|x\|=1\}$. Prove that, if $u\in C^1(\mathbb{R}^n)$, then: $$\lim_{\varepsilon\to 0}\int_{S(x_0,\varepsilon)}u(x) \operatorname{grad}(\varphi(x))\cdot N\,ds = u(x_0),$$
where $S(x_0,\varepsilon)=\{x\in \mathbb{R}^n:\|x-x_0\|=\varepsilon\}$ and $N$ is internal normal vector.
Where can I find a proof to this? or in which theorem should I look?