[Rafael-SOWNet]

This library's arccotangent is arctan(1/t), with range (-pi/2, pi/2]: arccot(-0.001) is -1.5698, arccot(0) is pi/2 and arccot(0.001) is 1.5698. It jumps at 0. A limit read the value at 0, so arccot(x) at 0 was pi/2 from the left, where it is -pi/2, and pi/2 two-sided, where there is no limit. Quotients over it inherited the wrong sign.

Where the argument arrives at 0, the arccotangent is now answered from the sign it arrives with: pi/2 from above, -pi/2 from below. A two-sided limit has a value only when both sides arrive with the same sign. Substitution and the paths that take a factor out no longer read the value at the jump. SitsOnAJump now names the arccotangent's jump next to sgn's.

Input, at 0	side	2.6.0	master	this branch
arccot(x)	left	pi/2	pi/2	-pi/2
arccot(x)	right	pi/2	pi/2	pi/2
arccot(x)	both	pi/2	pi/2	NaN
arccot(-x^2)	both	pi/2	pi/2	-pi/2
arccot(x) + arctan(x)	both	pi/2	pi/2	NaN
arccot(x) / x	left	-oo	-oo	+oo
arccot(x) / x	both	NaN	NaN	+oo
x * arccot(x) / sin(x)^2	both	NaN	NaN	+oo
(arccot(x) - pi/2) / x	both	-1	-1	NaN
(arccot(x) - pi/2) / x	right	-1	-1	-1
(arccot(x) + pi/2) / x	left	-oo	-oo	-1
arccot(x + 1)	both	pi/4	pi/4	pi/4
arccot(x * arccot(x))	left	pi/2	pi/2	pi/2
sgn(x * arccot(x))	left	unevaluated	-1	1
sgn(x * arccot(x))	both	unevaluated	NaN	1
floor(x * arccot(x))	left	unevaluated	-1	0
1 / (x * arccot(x))	left	-oo	-oo	unevaluated
arccot(x * sin(1/x))	both	pi/2	pi/2	unevaluated
arccot((x^2 - 1) - (x - 1)(x + 1))	both	pi/2	pi/2	pi/2

Each value on this branch follows from arccot(t) = pi/2 - t + O(t^3) for t -> 0+ and -pi/2 - t + O(t^3) for t -> 0-. The rows that changed are in BREAKING-CHANGES.md, and the new test pins them.

The sign of an expression that vanishes at the point was read from its first non-vanishing derivative there. Where that derivative holds arccot(x), putting 0 in reads the value at the jump, so x * arccot(x), which is positive on both sides of 0, came out negative from the left. On master that gave the wrong sgn and floor rows above. Once arccot was taken from the side, it also turned arccot(x * arccot(x)) into -pi/2. That reading now declines, and the sign is read from the factors instead. An argument that is 0 everywhere gets the constant pi/2. An independent review of this branch found these cases.

LimitTest.TestEquivalenceTableTo0 pinned (arccot(x) - pi/2) / x at 0, two-sided, as -1. That holds only for the textbook (0, pi) range. Under this library's range the left side is +oo, so the row moves to the new test as one-sided cases.

Checked on this branch:

• Tests: UnitTests 19348 passed; one improper-integral test hit its 5-minute wall-clock cap while the machine was loaded and passes alone in 27 s, plus the MCP, Terminal, InteractiveWrapper and F# wrapper suites, and the AOT smoke test.
• Harnesses: BoundCheck, RootCheck, CasBench, PropCheck, SimpSweep, RuleCheck, CanonCheck and Confluence all pass.
• limitcheck at DEPTH=3, COUNT=600, on 42 seeds against master: no answer changed. Three rows moved under load, on seeds 16, 31 and 77: one now runs past the 5 s cap and another no longer does. On a quiet machine each gives the same answer, in the same time, on master and on this branch.

🤖 Generated with Claude Code

[codecov]

Codecov Report

❌ Patch coverage is 74.07407% with 7 lines in your changes missing coverage. Please review.
✅ Project coverage is 86.80%. Comparing base (6d5f175) to head (447c0b4).
⚠️ Report is 1 commits behind head on master.

Files with missing lines	Patch %	Lines
...nctions/Continuous/Limits/Solvers/Limit.Classes.cs	73.33%	1 Missing and 3 partials ⚠️
...ath/Functions/Continuous/Limits/Transformations.cs	77.77%	0 Missing and 2 partials ⚠️
...nctions/Continuous/Limits/Solvers/Limit.Solvers.cs	66.66%	0 Missing and 1 partial ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #2420      +/-   ##
==========================================
- Coverage   86.82%   86.80%   -0.02%     
==========================================
  Files         312      312              
  Lines       69523    69545      +22     
  Branches    20305    20312       +7     
==========================================
+ Hits        60360    60366       +6     
- Misses       4930     4942      +12     
- Partials     4233     4237       +4     

Flag	Coverage Δ
cpp	37.77% <ø> (ø)
csharp	87.34% <74.07%> (-0.02%)	⬇️
fsharp	32.97% <ø> (ø)
interactive	39.81% <ø> (ø)

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Files with missing lines	Coverage Δ
...nctions/Continuous/Limits/Solvers/Limit.Solvers.cs	87.34% <66.66%> (-0.39%)	⬇️
...ath/Functions/Continuous/Limits/Transformations.cs	88.83% <77.77%> (-0.03%)	⬇️
...nctions/Continuous/Limits/Solvers/Limit.Classes.cs	76.95% <73.33%> (-0.49%)	⬇️

... and 10 files with indirect coverage changes

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