Yep, I’ve written numerical methods papers, and am very well aware of the field.

A limit process is a definition. Try computing with it. You’ll end up with an infinite sequence, or an approximation.

An iterative process is an infinite series. They’re equivalent.

Newtons method is the same. Completely equivalent to an infinite series as you increase precision.

And both require constants, infinitely precise. So you’re still not doing anything the 1/(x-y) operation cannot do, and to do those series you’ll compute using things amenable to being done via ops easy to do by hand or machine via the 1/(x-y) op.