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Mar 21, 2018 at 20:18 comment added Olivier Esser If it is undecidable whether the limit is 0; then it is undecidable whether the limit exists since obviously $\lim_{x\rightarrow 0} f(x)\neq 0$ iff $\lim_{x\rightarrow 0} {1\over f(x)}$ exists. But I agree this is a different (although interesting) problem.
Mar 21, 2018 at 12:23 comment added Alexandre Eremenko Yes, it is undecidable whether the limit is 0, and I suppose undecidable whether the limit exists. But this is a somewhat different problem: we want to know: IF the limit exists is it elementary.
Mar 20, 2018 at 0:54 history answered Igor Rivin CC BY-SA 3.0