I have added to my book section “Expressing limits as implications”.

The main (easy to prove) theorem basically states that $latex \lim_{x\to\alpha} f(x) = \beta$ when $latex x\to\alpha$ implies $latex f(x)\to\beta$. Here $latex x$ can be taken an arbitrary filter or just arbitrary ultrafilter.

The section also contains another, a little less obvious theorem. There is also a (seemingly easy) open problem there.