Proof for the following nascent delta function

Let $f(x)$ be a (nonzero) rapidly decreasing function in $\mathbb{R}$ (i.e. $f(x)$ and all its derivatives go to zero as $x\to\pm\infty$ faster than any negative power of $x$). The sequence of functions$$\eta_\epsilon(x)=\frac{\epsilon f(x)}{[\epsilon \pi f(x)]^2+x^2}$$approximates the delta distribution $\delta(x)=\lim_{\epsilon\to 0}\eta_\epsilon(x)$.

Any idea for proving of this?

It is similar to the well-known example $\frac{\epsilon}{\pi(\epsilon^2+x^2)}$, but not the same.