Answer the indeterminate forms that are not quotients

[Rafael-SOWNet]

Of the three indeterminate powers, only 1^oo had a rule — the second remarkable limit. 0^0 and oo^0 reached the descent, which substitutes each part's own limit and hands back 0^0, that is NaN: the claim that the limit does not exist. A product of something vanishing with something diverging had no reading either, unless one of its factors happened to be written as a reciprocal.

	before	after
lim x->0+ x^sin(x)	NaN	1
lim x->0+ sin(x)^x	NaN	1
lim x->0+ x^(1/ln(x))	NaN	e
lim x->0+ x^(2/ln(x))	NaN	e²
lim x->+oo (1/x)^(1/x)	NaN	1
lim x->0+ sin(x)·ln(x)	NaN	0
lim x->0+ sqrt(x)·ln(x)	NaN	0
lim x->0+ x·cotan(x)	NaN	1
lim x->0+ (1-cos(x))·cotan(x)	NaN	0

19 of 20 measured cases against 9 before. x^(1/ln(x)) is the one worth naming: 0^0 is not always 1, which is the whole reason it is indeterminate, and here the exponent decides on e.

How

Both families go over to the textbook form — e^(g·ln f) for the power, and the diverging factor over the reciprocal of the vanishing one for the product.

The exponent is asked as a limit of its own rather than rewritten in place. A rewrite would only hand the descent a product it reads no better than the power, because the descent substitutes each part's limit and does not apply l'Hopital's rule to a part. Asking outright is what puts the whole machinery behind it — my first attempt did rewrite in place, and it regressed lim x->+oo x^(1/x) from 1 to NaN.

Two guards, both from measurement

The reciprocal split keeps precedence, but only when it produces something usable. x·e^(-x) splits into the clean x/e^x, and inverting the other half instead does not terminate — so the split goes first. But it can also hide the indeterminacy it was meant to expose: tan(x)·ln(x) has been written sin(x)/cos(x)·ln(x) by the time it arrives, and splitting gives sin(x)·ln(x)/cos(x), whose divisor tends to 1. So the split is used only when what it produces is genuinely 0/0 or oo/oo.

A base the library cannot differentiate is declined. Every route out of ln(f) runs through differentiating f, and Factorialf.InnerDifferentiate returns NaN outright — the digamma function is not implemented. Without this guard lim x->+oo ((x!)/x^x)^(1/x) had no answer either way and took twenty seconds not to find one; it showed up as a timeout in a 117-problem corpus that had been at zero all along. Caught before this was opened, not after.

What is deliberately unchanged

lim x->0 x^x still has no two-sided limit, and that is right rather than a gap. x^x is not real to the left of 0, so the 1 that comes back from that side is the complex continuation and not a limit for the right-hand one to agree with. Limits.TestNoLimit has pinned this all along; an earlier draft of mine made the two-sided case answer 1 and broke that test, which was the test being correct and me being wrong.

tan(x)·ln(x) at 0+ is the one measured case still missing. It arrives restructured as a Divf rather than a Mulf, so the product reading never sees it. It is NaN on master too, so this is a gap rather than a regression, and the fix belongs with whatever handles the restructuring.

Measurements

• 25 new tests.
• Suite: Failed: 0, Passed: 4597, Skipped: 14, Total: 4611, plus 127 F#.
• 117-problem corpus: 111/117, 0 wrong / 0 error / 0 timeout.

Independent of #703 and #709.

[codecov]

Codecov Report

❌ Patch coverage is 81.81818% with 10 lines in your changes missing coverage. Please review.
✅ Project coverage is 81.82%. Comparing base (90c00a8) to head (1b1eb9b).
⚠️ Report is 82 commits behind head on master.

Files with missing lines	Patch %	Lines
...ath/Functions/Continuous/Limits/Transformations.cs	84.31%	4 Missing and 4 partials ⚠️
...ns/Continuous/Limits/Solvers/Solvers.Definition.cs	50.00%	1 Missing and 1 partial ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #710      +/-   ##
==========================================
+ Coverage   80.99%   81.82%   +0.82%     
==========================================
  Files         155      159       +4     
  Lines       13687    13854     +167     
  Branches     1957     2345     +388     
==========================================
+ Hits        11086    11336     +250     
+ Misses       1990     1861     -129     
- Partials      611      657      +46     

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