The command RationalFunctionLimit(f,p) returns either undefined if the rational function f does not admit a finite limit at the point given by p, or the limit of f at p otherwise.

If p is a pole of f, that is, if p cancels the denominator of f, then it is assumed that p is an isolated pole of f, that is, f has no poles other than p in a neighborhood of p.

This command is part of the RegularChains[AlgebraicGeometryTools] package, so it can be used in the form RationalFunctionLimit(..) only after executing the command with(RegularChains[AlgebraicGeometryTools]).  However, it can always be accessed through the long form of the command by using RegularChains[AlgebraicGeometryTools][RationalFunctionLimit](..).

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$$\mathrm{with}\left(\mathrm{AlgebraicGeometryTools}\right)\:$

$\mathrm{RationalFunctionLimit}\left(\frac{x^{2}y{z}^2}{x^4\+y^4\+z^4}\,\left[x\=0\,y\=0\,z\=0\right]\right)$

$0$

$\mathrm{RationalFunctionLimit}\left(\frac{wz\+x^2\+y^2}{w^2\+x^2\+y^2\+z^2}\,\left[x\=0\,y\=0\,z\=0\,w\=0\right]\right)$

$\mathrm{undefined}$

$\mathrm{RationalFunctionLimit}\left(\frac{x^6}{w^6\+l^2\+t^2\+x^2\+y^2\+z^2}\,\left[x\=0\,y\=0\,z\=0\,w\=0\,t\=0\,l\=0\right]\right)$

$0$

Parisa Alvandi, Changbo Chen, Marc Moreno Maza "Computing the Limit Points of the Quasi-component of a Regular Chain in Dimension One." Computer Algebra in Scientific Computing (CASC), Lecture Notes in Computer Science - 8136, (2013): 30-45.

Parisa Alvandi, Masoud Ataei, Mahsa Kazemi, Marc Moreno Maza "On the Extended Hensel Construction and its application to the computation of real limit points." J. Symb. Comput. 98: 120-162 (2020)

The RegularChains[AlgebraicGeometryTools][RationalFunctionLimit] command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.