quantum mechanics - Physical position eigenfunction normalisation

We know that the Dirac function $$\delta(a)=\lim_{a \rightarrow 0} \delta_{a}(x)$$ can be written as an infinitesimally narrow Gaussian: $$\delta_{a}(x) := \frac{1}{\sqrt{2\pi a^2}}e^{-x^2/2a^2}$$

Our professor told us that for any value $a>0$, the physical position eigenfunction is $$\psi_{x_0}(x)\cong N_1\delta_a(x-x_0).$$

How can I show that $\psi_{x_0}$ is a physical position eigenfunction?