Let's say that a function $d:S\times S\to [0,\infty)$ for a countable set $S$ is a metric in the limit if $$d(x,y)\le \liminf_{n\to\infty} d(x,z_n)+d(z_n,y),$$ $$\lim_{n\to\infty} d(z_n,z_n)=0, \quad\text{and}\quad \lim_{n\to\infty} d(x,z_n)- d(z_n,x)=0$$ for any sequence $z_n$ that goes to infinity (i.e., is eventually not equal to any particular point). We could require that $d(x,y)=0\implies x=y$ as well.

I'm looking for an answer to any one of these:

1. Is this a familiar idea?
2. Does some of the theory of metric spaces still hold for such functions?
3. Perhaps there exists a "better" variant of this idea?