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What algorithm does Mathematica use to compute limits for functions in $\mathbb{R}^n$ with $n\geq2$? Here I refer to limits that present indeterminate forms. How can it test every possible path? Does it test a couple and then if they are all equal use epsilon-delta proof? If so, how would this work for Mathematica? Does it do it numerically?

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    $\begingroup$ (1) It is a melange of methods, a poly-algorithm at her Polly finest. $\endgroup$ – Daniel Lichtblau Feb 27 '18 at 22:36
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    $\begingroup$ (2) For rational functions, and frequently for more general meromorphic functions, there are ways to reduce to a finite set of paths. It uses some heuristics to decide when one or another tactic might be more efficient. $\endgroup$ – Daniel Lichtblau Feb 27 '18 at 22:38
  • $\begingroup$ Thanks for the reply. Is there a source/s I can look at so I learn more about this? Thank you. $\endgroup$ – user372003 Feb 28 '18 at 7:28
  • $\begingroup$ Related: mathematica.stackexchange.com/q/21544 but perhaps some progress has been made since then. $\endgroup$ – Michael E2 Feb 28 '18 at 13:12
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    $\begingroup$ One approach, so to speak, can be found in the ISSAC 2016 paper "Computing Limits of Real Multivariate Rational Functions" by Alvandi, Kazemi, and Moreno Maza. It is not using quite the same method as those in Mathematica 11.2 but there is some similarity. I should mention that I did not write the code so I am not familiar with all the details under the hood. $\endgroup$ – Daniel Lichtblau Feb 28 '18 at 16:03

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