[Rafael-SOWNet]

x / (tanh(1/2) - tan(x)) at 1 took 10 s, and tanh(x) / (tanh(1/2) - tan(x)) at 1 took 18 s, at points where both are continuous. limitcheck at SEED=28 had thirteen limits that did not finish in 5 s, and twelve of them were these.

A limit at a finite point moves the variable out to infinity, x = a ± 1/x. It then simplified the whole expression before reading its value there. That value is what the substitution answers a continuous point with in any case, under the same guards for a jump, a cut and an indeterminate power, and simplifying was the expensive half.

The substitution is now also asked first, and it answers where its value is a finite real. It stays out of complex values, since a complex value can be the value on a cut, and the cut guards were written against the simplified form. One difference of form: a tangent can come back as the quotient of its sine and cosine.

One guard was owed. Reading the value first hands the descent an argument's limit already folded, -1 rather than -1 - 1/+oo. The factorial's descent put that straight into x! and wrote down the pole, so x! at -1 came back (-1)!. The factorial now declines at a negative integer.

limitcheck, DEPTH=3, seeds 14, 16, 19, 23, 26, 28 and 29, against master f3e514a:

	master	this branch
total time	667 s	437 s
the limit did not finish in 5 s	13	1
compared	36 275	36 294
the limit is another value	7	7
the limit is NaN where the values settle	84	85

Read the last row.

• Two wrong NaNs are gone. abs(x + 1 - x^x) at 0 from each side is now unevaluated.
• Three limits that were unevaluated are now NaN: sin(x)^(log(x, pi)^arcsin(x)) at 1, (2(x - pi))^ln(-x) at 0, and arctan((-1)^x)^(pi/x/cos(2)) at 1/2. Each has no real values on a side: (-oo)^(pi/2) from the left, ln(-x) for x > 0, and (-1)^x throughout. A side with no real values making no two-sided limit is the convention x^x at 0 is pinned to, by LimitTest.TestNoLimit, and limitcheck flags x^x the same way in every run.

I tried making the two-sided path claim NaN only where both sides are decided in full. That took these three back to unevaluated, but it also unpinned x^x, x ln x and sin(x) ln x. That is a decision about the convention and not a side effect to slip into a speed fix, so it is not here.

Every changed answer is recorded in BREAKING-CHANGES.md. Suite, AOT and the harnesses are green locally.

Found by limitcheck at SEED=28; no issue was filed.

🤖 Generated with Claude Code

[codecov]

Codecov Report

❌ Patch coverage is 81.81818% with 2 lines in your changes missing coverage. Please review.
✅ Project coverage is 86.78%. Comparing base (f3e514a) to head (8e30927).

Files with missing lines	Patch %	Lines
...nctions/Continuous/Limits/Solvers/Limit.Classes.cs	66.66%	0 Missing and 2 partials ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #2365      +/-   ##
==========================================
+ Coverage   86.77%   86.78%   +0.01%     
==========================================
  Files         312      312              
  Lines       68855    68861       +6     
  Branches    20063    20066       +3     
==========================================
+ Hits        59746    59762      +16     
+ Misses       4923     4909      -14     
- Partials     4186     4190       +4     

Flag	Coverage Δ
cpp	37.77% <ø> (ø)
csharp	87.32% <81.81%> (+0.01%)	⬆️
fsharp	32.97% <ø> (ø)
interactive	40.72% <ø> (ø)

Flags with carried forward coverage won't be shown. Click here to find out more.

Files with missing lines	Coverage Δ
...ns/Continuous/Limits/Solvers/Solvers.Definition.cs	89.86% <100.00%> (+0.19%)	⬆️
...nctions/Continuous/Limits/Solvers/Limit.Classes.cs	73.38% <66.66%> (-1.99%)	⬇️

... and 6 files with indirect coverage changes

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