The meaning of infinity in calculus and computer algebra systems

We use filters of open sets to provide a semantics justifying formally the use of infinity in informal limit calculations in calculus, and in the same kind of calculations in computer algebra. We compare the behavior of these filters to the way Mathematica behaves when calculating with infinity. A proper semantics for computer algebra expressions is necessary not only for the correct application of those methods, but also in order to use results and methods from computer algebra in theorem provers. The computer algebra method under discussion in this paper is the use of rewrite rules to evaluate limits involving infinity. Unlike in other areas of computer algebra, where the problem has been a mismatch between a known semantics and implementations, we here provide the first precise semantics.