integration – Relation between the limit of a function and its exponential integral

I am working reading a textbook on scattering theory where they state the following relation, $$lim_{trightarrow -infty} f(t) = lim_{epsilon rightarrow 0} epsilon int_{-infty}^{0} e^{epsilon t} f(t).$$
This equation should hold for any function $f(t)$ for which the limit on the lhs exists. I have googled a lot but I can not find any mention of this equation anywhere else. Does anyone know what theorem relates to this equation or what steps I could take to attempt a proof. Even intuitively this equation makes little sense to me, but indeed with any test function I try it does work.

Thanks!