I have to compute the following integral
$$\lim_{\epsilon \rightarrow 0} \int_a^\infty \delta_\epsilon (x) f(x) dx$$
where $\delta_\epsilon$(x) is such that $\int_{-\infty}^\infty \delta_\epsilon (x) dx=1$, for every $\epsilon$, $f$ is an arbitrary function and
$$\lim_{\varepsilon\rightarrow 0}\delta_\varepsilon(x)=\begin{cases} 0 & \text{if } x\neq0 \\ \infty & \text{if } x=0 \end{cases}$$
In other words, the limit of $\delta_\varepsilon$ is the delta dirac function. My guess is that the integral is given by:
$$\lim_{\epsilon \rightarrow 0} \int_a^\infty \delta_\epsilon (x) f(x) dx = \begin{cases} f(0) & \text{if } a<0 \\ 0 & \text{otherwise} \end{cases}$$
But I am not sure. Intuitively, it seems correct to me, but I could not find this result stated formally anywhere.