Experimenting with Apéry Limits and WZ pairs

This article, dedicated with admiration in memory of Jon and Peter Borwein,illustrates by example, the power experimental mathematics, so dear to them both, experimenting so-called Apéry limits WZ pairs. In particular we prove a weaker form an intriguing conjecture Marc Chamberland Armin Straub (in article Borwein), generate lots new limits. We also rediscovered infinite family cubic irrationalities, that suggested very good effective irrationalitymeasures (lower than Liouville's generic 3), conjectured go down optimal 2. As it turned out, as pointed out Paul Voutier (see postscript kindly written him), our conjectures follow from deep results number theory. Nevertheless believe further experiments Maple programs would lead interesting results.