In nonstandard analysis, we have hyperreal numbers that are greater than any real number. As such, we can create a sequence of infinite, hyperfinite hyperreal numbers which grows ever smaller.
More specifically if we can find such a sequence $(x_n)_{n\in M}$ such that for every hyperreal number $h\in{^*\mathbb R}$ we have $x_i <h$ for some $i\in M$, this sequence feels like a general way of defining a limit that runs against infinity, but from above.
I.e. if $f$ is a function, I am interested in $\lim_{n\to\infty} f(x_n)$.
I've named the index of the sequence $M$ because I'm not sure how $M$ should look like.
The concept seems interesting enough that I hope that somebody may have taken the time to find out whether it can be made precise in a meaningful way.
As such I'm looking for papers or similar which investigate this concept, and especially if it is even possible to define the limit from above in a meaningful way.