Algorithms for Calculating Limits and Asymptotic Forms

The algorithms used in symbolic-computation systems for calculating asymptotic forms of functions are generally based on some kind of series expansion. However the computation may fail to terminate when terms repeatedly cancel. A simple but instructive example is given by the function \[\exp(x^{-1} + e^{-x}) - \exp(x^{-1}).\] If the outermost exponentials are expanded naively the powers of $x^{-1}$ will dominate but will repeatedly cancel. We look at the ways in which some recent algorithms have attacked this problem.