[Rafael-SOWNet]

A follow-up to #2358. No issue was filed.

at 0	2.6.0 and master	now
cos(x)^(1/x^2)	NaN	e ^ (-1/2)
limitright(cos(x)^(1/x^2), x, 0)	unevaluated	e ^ (-1/2)
(sin(x)/x)^(1/x^2)	NaN	e ^ (-1/6)
cos(2x)^(1/x^2)	NaN	e ^ (-2)
(1 + sin(x))^(1/x)	NaN	e
limitright(cos(x)^(1/x), x, 0)	unevaluated	1
(1 + x)^(1/x), (1 - x^2)^(1/x^2)	e, 1/e	unchanged
x * ln(x), sin(x) * ln(x), x^x two-sided	NaN	unchanged

The second remarkable limit writes a 1^oo power f^g as e^(g (f - 1)). The descent reads the product in its exponent as 0·∞, NaN, the claim that there is no limit. Only a base of 1 plus a polynomial reached an answer, by another route. The exponent is now the quotient (f - 1) / (1/g), a 0/0 that #2358 asks whole, and it has its value. Both places the rule builds the power are changed.

Why a quotient and not a product. The first version gave a product in the descent the same whole-limit question as #2358's quotients. It answered all of these, and also answered x * ln(x) at 0 two-sided with 0, sin(x) * ln(x) too. Those are pinned NaN, because the two-sided reading keeps a limit whose left side is not real out of the reals (RealCodomainLimitTest, FirstRemarkableOverFactorsTest). The two-sided reading asks one-sided sub-limits, and those had relied on 0·∞ being NaN. Restricting the product retry to one-sided limits did not separate the two, so it was dropped. The quotient changes only the exponent the rule builds.

limitcheck at SEED=14 and 16, and at DEPTH=3 on 19 and 23, matches master exactly; none of those samples is a 1^∞ power. 4194 limit, integration and remarkable-limit tests pass.

Tests: AOneToTheInfinityLimitThroughItsExponentTest, 11 cases. 6 fail on master; the 2 controls and the 3 two-sided conventions pass on both. The full suite, AOT and the nine harnesses pass. There is a BREAKING-CHANGES entry, measured on 2.6.0.

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[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 86.77%. Comparing base (553e68d) to head (e2d60d8).

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #2359      +/-   ##
==========================================
+ Coverage   86.76%   86.77%   +0.01%     
==========================================
  Files         312      312              
  Lines       68807    68809       +2     
  Branches    20045    20045              
==========================================
+ Hits        59697    59712      +15     
+ Misses       4924     4911      -13     
  Partials     4186     4186              

Flag	Coverage Δ
cpp	37.77% <ø> (ø)
csharp	87.32% <100.00%> (+0.01%)	⬆️
fsharp	32.97% <ø> (ø)
interactive	40.72% <ø> (ø)

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Files with missing lines	Coverage Δ
...ath/Functions/Continuous/Limits/Transformations.cs	87.86% <100.00%> (+0.04%)	⬆️

... and 6 files with indirect coverage changes

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