Computing limits with the regularchains and powerseries libraries: from rational functions to Zariski closure

Many fundamental concepts in mathematics are defined in terms of limits and it is desirable for computer algebra systems to be able to compute them. However, limits of functions, limits of secants or topological closures are, by essence, hard to compute in an algorithmic fashion, say by doing finitely many rational operations on polynomials or matrices over the usual coefficient fields of symbolic computation. This is why a computer algebra system like Maple is not capable of computing limits of rational functions in more than two variables while it can perform highly sophisticated algebraic computations like solving (formally) a system of partial differential equations.