I need to show $ \int_{S^{n-1}} \frac{1}{\sqrt\epsilon} e^{-\frac{(\omega,x)^2}{2\epsilon}} d\omega \to \frac{C}{|x|}$ as $ \epsilon \to 0 $, where $ (\centerdot,\centerdot) $ denotes the dot product in $ \mathbb R^n $ and $ C $ is some constant, in the sense of distributions.
I assume it is known that $ \frac{1}{\sqrt\epsilon}e^{-\frac{x^2}{2\epsilon}} \to \delta $ as $ \epsilon \to 0 $ in the sense of distributions. Also I know that in order to show the convergence of a sequence of generalized functions $ \{f_n\} $ to a limiting function $ f $, I need to verify that $ \lim_{n \to \infty}\langle f_n,\phi \rangle = \langle f,\phi \rangle $ for all test functions $\phi \in \mathscr D(\mathbb R^n)$.
I have tried to see what I can do about this integral:
$$ \int_{\mathbb R^n} \left( \int_{S^{n-1}} \frac{1}{\sqrt\epsilon} e^{-\frac{(\omega,x)^2}{2\epsilon}} d\omega \right) \phi(x) \ dx $$
I made a polar change of variables in the exterior integral, but, unfortunately, it did not lead to anything reasonable. I need to figure out this convergence within the larger goal of finding the Fourier Transform of the function $ 1/|x|^{n-1} $ as a temperate distribution.