An infinity over a vanishing divisor is the infinity of the signs

BREAKING-CHANGES.md

@@ -85,7 +85,7 @@ and some are left unevaluated ([#2267](https://github.com/asc-community/AngouriM
 |---|---|---|
 | `"limitright(ln(x) / ln(cos(x)), x, 0)".ToEntity().Simplify()` | `-oo` | `+oo` |
 | `"limitright(ln(sin(x)) / ln(cos(x)), x, 0)".ToEntity().Simplify()` | `-oo` | `+oo` |
-| `"limit(ln(x^2) / ln(cos(x)), x, 0)".ToEntity().Simplify()` | `-oo` | `NaN`, which is wrong too: both sides are `+oo` ([#2274](https://github.com/asc-community/AngouriMath/issues/2274)) |
+| `"limit(ln(x^2) / ln(cos(x)), x, 0)".ToEntity().Simplify()` | `-oo` | `+oo`, by [#2274](https://github.com/asc-community/AngouriMath/issues/2274) |
 
 ### A value put in for a derivative's variable is where it is evaluated
 
@@ -544,6 +544,20 @@ the expression's own domain as its condition ([#2142](https://github.com/asc-com
 | `"(1 - cos(a))/(1 + cos(a))/tan(a/2)^2".Simplify()` | `(1 - cos(a)) / ((cos(a) + 1) * tan(a / 2) ^ 2)` | `1 provided not cos(a) + 1 = 0 and not tan(a / 2) = 0 and not cos(a / 2) = 0` |
 | `"(cos(2*x) - 1)^2 + sin(2*x)^2 - 4*sin(x)^2".Simplify()` ([#2133](https://github.com/asc-community/AngouriMath/issues/2133)) | `2 + (-2) * cos(2 * x) + (-4) * sin(x) ^ 2` | `0` |
 
+### An infinity over a vanishing divisor is the infinity of the signs
+
+**Silent: a limit of a quotient whose dividend tends to an infinity and whose divisor vanishes from one side is that
+infinity times the divisor's sign, where a two-sided one was `NaN`.** The quotient of the parts' limits, `(-oo) / 0`, says
+nothing about the side the divisor vanishes from, and the rule that reads the side asked for a finite dividend. So each
+side of `ln(x^2) / (cos(x) - 1)` at 0 came back undecided, and the two-sided limit, `+oo` from both, was `NaN`
+([#2274](https://github.com/asc-community/AngouriMath/issues/2274)).
+
+| Input | Was (2.6.0) | Now |
+|---|---|---|
+| `"limit(ln(x^2) / (cos(x) - 1), x, 0)".ToEntity().Simplify()` | `NaN` | `+oo` |
+| `"limit(ln(x^2) / ln(1 - x^2/2), x, 0)".ToEntity().Simplify()` | `NaN` | `+oo` |
+| `"limit(ln(abs(x)) / ln(cos(x)), x, 0)".ToEntity().Simplify()` | `NaN` | `+oo` |
+
 ### The double factorial is the postfix `n!!`
 
 **Not silent: `n!!` is now the double factorial, where it was a parse error.** It is the product of the positive integers

Sources/.editorconfig

@@ -185,6 +185,9 @@ file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed
 [Tests/UnitTests/Calculus/BoundNameTest.cs]
 file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n
 
+[Tests/UnitTests/Calculus/AnInfinityOverAVanishingDivisorTest.cs]
+file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n
+
 [Tests/UnitTests/Core/BinderShadowingTest.cs]
 file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n
 

Sources/AngouriMath/Functions/Continuous/Limits/Transformations.cs

@@ -1056,9 +1056,11 @@ internal static bool IsDeterminate(Entity combined, Variable x)
             if (EvalAssumingContinuous(divisor.Limit(x, dest, side)) is not Real { IsZero: true })
                 return null;
             // A dividend that vanishes too makes the quotient indeterminate rather than
-            // divergent, and one that diverges is a different question again. Only a dividend
-            // with a definite non-zero size leaves the divisor to decide the answer.
-            if (EvalAssumingContinuous(dividend.Limit(x, dest, side)) is not Real { IsFinite: true, IsZero: false } dividendLimit)
+            // divergent. One with a definite sign that does not vanish, a finite one or an
+            // infinity, leaves the divisor to decide the answer: -oo over a divisor vanishing from
+            // below is +oo, and ln(x^2) / (cos(x) - 1) at 0 was left as (-oo) / 0, which is NaN.
+            // https://github.com/asc-community/AngouriMath/issues/2274
+            if (EvalAssumingContinuous(dividend.Limit(x, dest, side)) is not Real { IsZero: false, IsNaN: false } dividendLimit)
                 return null;
             if (SignFromVanishingOrder(divisor, x, dest, side) is not { } divisorSign)
                 return null;

Sources/Tests/UnitTests/Calculus/AnInfinityOverAVanishingDivisorTest.cs

@@ -0,0 +1,43 @@
+//
+// Copyright (c) 2019-2026 Angouri.
+// AngouriMath is licensed under MIT.
+// Details: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.
+// Website: https://am.angouri.org.
+//
+
+using AngouriMath.Extensions;
+using Xunit;
+
+namespace AngouriMath.Tests.Calculus
+{
+    /// <summary>
+    /// A quotient whose dividend tends to an infinity and whose divisor vanishes from one side tends to the infinity of
+    /// the two signs' product. The quotient of the parts' limits, (-oo) / 0, says nothing about the divisor's side and is
+    /// NaN, and the rule that reads the side asked for a finite dividend: ln(x^2) / (cos(x) - 1) at 0, +oo from both
+    /// sides, was NaN, the claim that it has no limit.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/2274">#2274</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AnInfinityOverAVanishingDivisorTest
+    {
+        [Theory]
+        [InlineData("limit(ln(x^2) / ln(1 - x^2/2), x, 0)", "+oo")]
+        [InlineData("limit(ln(x^2) / (cos(x) - 1), x, 0)", "+oo")]
+        [InlineData("limit(ln(abs(x)) / ln(cos(x)), x, 0)", "+oo")]
+        [InlineData("limit(ln(x^2) / ln(cos(x)), x, 0)", "+oo")]
+        [InlineData("limitright(ln(x) / sin(x), x, 0)", "-oo")]
+        [InlineData("limitleft(ln(-x) / sin(x), x, 0)", "+oo")]
+        [InlineData("limitright((1/x) / sin(x), x, 0)", "+oo")]
+        [InlineData("limit(1/x^2, x, 0)", "+oo")]
+        public void ItIsTheInfinityOfTheSigns(string limit, string value)
+            => Assert.Equal(value.ToEntity(), limit.ToEntity().Simplify());
+
+        /// <summary>Where the sides disagree, or one has no real value, there is still no limit.</summary>
+        [Theory]
+        [InlineData("limit(1/x, x, 0)")]
+        [InlineData("limit(x^x, x, 0)")]
+        [InlineData("limit(ln(x) / x, x, 0)")]
+        public void WhereTheSidesDoNotAgreeThereIsNone(string limit)
+            => Assert.Equal(MathS.NaN, limit.ToEntity().Simplify());
+    }
+}