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we need more limit solvers #231
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@MomoDeve 2 out of 3 are closed. But it seemingly does not recognize functions of higher order.
All three cases on the list now answer, including the one still unticked:
lim x->+oo (1 + 1/x)^xelim x->0 sin(x)/x1lim x->+oo (e^x - x)+ooThe last came from #682, which reads a difference of two divergent parts through the ratio of its halves, and #694, which added Gruntz's algorithm behind it.
A fair amount else landed alongside: l'Hopital's rule at infinity (#680), the one-sided path getting everything the two-sided path had (#697, #699, #700), and the indeterminate forms that are not quotients —
0^0,oo^0, and products of something vanishing with something diverging (#710).Leaving this open rather than closing it, since it says more cases will be added here and is useful as a running list. If you would rather it closed now that its three are done, say so. If you have further cases, add them and I will take them.
This is closable if research (see Mathematica, sympy etc) shows that there are no more classes of limits that an extra solver can solve.
Taking that as the standard to meet. Recording where it stands so the research has a starting point rather than being redone from scratch.
The three classes on the original list all answer, and since then the machinery has grown: Gruntz's MRV algorithm (#694), l'Hopital applied one-sidedly, indeterminate powers via the exponent's own limit, vanishing-times-diverging products, equivalent infinitesimals, and as of #714 piecewise limits and determinate recombination of parts.
What a comparison against sympy and Mathematica should be looking for, from what has actually come up here:
- Asymptotic expansions of the special functions.
lim x->+oo ((x!)/x^x)^(1/x)= 1/e is declined today because differentiating a factorial gives NaN. This is the one concrete gap I can name, and it is Stirling — sympy answers it viagammasimp/series, not via a limit rule. - Limits along a direction in the complex plane.
ApproachFromis Left/Right/BothSides, so there is no way to ask for a limit alonge^(i*theta). Raised on Unexpected behavior of limits #596 too. - Limits of sequences as distinct from functions —
lim n->oowhere n is an integer, which is where sums and recurrences would need it. - Oscillatory non-existence.
lim x->+oo sin(x)should be a definite "does not exist"; worth checking it is not merely unevaluated, since the two are different claims and this library is careful about that distinction elsewhere.
I have not done the sympy/Mathematica sweep itself, so I am not closing this on my own say-so — but if that list comes back empty apart from the first item, this is closable and the factorial asymptotic is its own issue.
- Asymptotic expansions of the special functions.
The one unticked case now works, on
masterata45a7256:lim(x -> +oo) e^x - x -> +ooSo all three boxes are done — the two already ticked and this one. The Gruntz machinery in
Functions/Continuous/Limits/Gruntz/is what answers it, which is also what #353 asks for.Looks closeable, unless you want it kept as a running list for cases added later — the body says "more cases will be added here", and a permanently-open bucket is a legitimate thing to have as long as it is understood to be one.
- added a commit that references this issue
on Sep 1, 2026 - added a commit that references this issue
on Sep 30, 2026
here are some missing cases:
lim(x -> inf) (1 + 1/x)^xlim(x-> 0) sin(x) / xlim(x -> inf) e^x - xmore cases will be added here