One of the most useful techniques I've ever learned is to use dual numbers to see what happens to some function $f(x)$ as $x \to $ some $r$. I will just evaluate $f(r+\epsilon)$ and multiply things out, and the automatic differentiation property of the dual numbers often ends up performing L'Hôpital's rule and etc for me. I have used this technique so many times I couldn't begin to count them all; I wish I'd learned about dual numbers and automatic differentiation much earlier.
I wish I had a similar technique to evaluate things as $x \to \infty$ this way.
I've often gotten this to work by simply treating $1/\epsilon$ as a formal quantity and just ignoring the fact that $1/\epsilon$ doesn't actually exist. Sometimes, doing that, it's possible to rewrite things in a way that these $1/\epsilon$ terms cancel out, or can otherwise be rewritten in terms of just regular $\epsilon$ terms, and then I can proceed with the automatic differentiation as usual. Having to do this is very tedious, however, and I hate it.
What I'm really looking for is something just as simple as the dual numbers which will do all of this for me. Does there exist some extremely simple algebraic structure that gives us some easy way to represent infinite quantities instead of infinitesimal ones?
Maybe a different real algebra, or maybe some kind of "extended real algebra" with elements squaring to $\infty$, or just something? Is it even possible to do this in a way that we get something associative?