I am trying to simplify the following limit of integral where $\mu$ is given:

$$p(y) = \lim_{\sigma \to 0} \int_{\mathbb R} |x| \cdot \frac{1}{\sqrt{2\pi\sigma^2} } e^{-\frac{1}{2\sigma^2} (xy - \mu)^2} f(x) dx,$$

however I am not sure if there is a way to simplify it, as the integrand does not converge under the limit $\sigma \to 0$ and the interchangeablity of limit and integral fails here:

$$\lim_{\sigma \to 0} |x| \cdot \frac{1}{\sqrt{2\pi\sigma^2} } e^{-\frac{1}{2\sigma^2} (xy - \mu)^2} f(x) = \begin{cases} \inf, & x \ne \mu /y; \\ 0, & x = \mu /y \end{cases}.$$

However, I DO know the answer of the limit of the integral, as naturally it is the probablity density function of $Y = T/X$, where $X$ has PDF $f(x)$ and $T \sim N(\mu, \sigma^2)$ is a normal distribution. The limit of integral at $\sigma \to 0$ then essentially means the PDF of $\mu / X$ and can be easily calculated.

So how do I calculate this limit of integral analytically, without the help of the intuition from probablity?