Read an even power of a real base as never negative in a limit

BREAKING-CHANGES.md

@@ -227,6 +227,23 @@ arctangent of something real, and an arcsine and an arccosine inside (-1, 1). Wh
 | `"sqrt(x^2)^(-3/2)".ToEntity().Limit("x", 0)` | `NaN` | `+oo` |
 | `"sin(arcsin(x)) * (tan(-1) + 2/x)".ToEntity().Limit("x", 0)` | `NaN` | `2` |
 
+### A limit over a vanishing even power is not claimed not to exist
+
+**Silent: `1/((x^2)^tanh(1))` at 0 is `+oo`, where it was `NaN`, the claim that the limit does not exist.** A base
+was read as never negative only for an absolute value, or a real power, product or quotient of never-negative parts,
+so `x^2` was not. A vanishing power of it to a constant that is not a number, or to an exponent tending to a negative
+number, was left unevaluated from each side, and the two-sided limit was `NaN`. An even power of a base real for
+real `x` is now never negative. `1/(((i*x)^2)^tanh(1))`, whose base is `-x^2`, is still not answered.
+
+| Input | Was (2.6.0) | Now |
+|---|---|---|
+| `"1/((x^2)^tanh(1))".ToEntity().Limit("x", 0)` | `NaN` | `+oo` |
+| `"1/((x^2)^tanh(1))".ToEntity().Limit("x", 0, ApproachFrom.Right)` | `limitright(1 / (x ^ 2) ^ ((e ^ (2 * 1) - 1) / (e ^ (2 * 1) + 1)), x, 0)` | `+oo` |
+| `"(sinh(x)^2)^(sin(x) - 1)".ToEntity().Limit("x", 0)` | `NaN` | `+oo` |
+| `"(sinh(x)^2)^(sin(x) - 1)".ToEntity().Limit("x", 0, ApproachFrom.Right)` | `limitright((((e ^ x - e ^ (-x)) / 2) ^ 2) ^ (sin(x) - 1), x, 0)` | `+oo` |
+| `"sqrt(x^2)^tan(pi - 1)".ToEntity().Limit("x", 0)` | `NaN` | `+oo` |
+| `"sqrt(x^2)^tan(pi - 1)".ToEntity().Limit("x", 0, ApproachFrom.Right)` | `limitright((sqrt(x ^ 2)) ^ tan(pi - 1), x, 0)` | `+oo` |
+
 ### An equation in two different square roots is solved exactly
 
 **Silent: an equation `m_1 sqrt(A) + m_2 sqrt(B) + P = 0` in x alone, with two different square roots, is answered in

Sources/AngouriMath/Functions/Continuous/Limits/Solvers/Limit.Classes.cs

@@ -333,7 +333,7 @@ partial record Powf
                 // was left unevaluated from each side for it. Where the base vanishes, it does from
                 // above.
                 if (baseLimit?.Evaled is Real { IsZero: true } && exponentLimit?.Evaled is Real { IsNegative: true, IsFinite: true }
-                    && IsNeverNegative(Base))
+                    && IsNeverNegative(Base, x))
                     return Real.PositiveInfinity;
                 var substituted =
                     (baseLimit, exponentLimit)

Sources/AngouriMath/Functions/Continuous/Limits/Solvers/Solvers.Definition.cs

@@ -249,7 +249,7 @@ bool RealThere(Entity node)
             // reads off the form, an even power of something real, and a real power that is itself
             // real -- sqrt(x^2), whose base vanishes at 0 and is not positive there.
             bool NotNegativeThere(Entity node)
-                => IsNeverNegative(node)
+                => IsNeverNegative(node, x)
                    || node is Powf(var @base, Integer { EInteger.IsEven: true }) && RealThere(@base)
                    || node is Powf(_, not Integer) && RealThere(node);
             bool StrictlyInsideTheUnitInterval(Entity node)

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

@@ -1102,7 +1102,7 @@ internal static bool IsDeterminate(Entity combined, Variable x)
             // 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 ((IsNeverNegative(divisor) ? 1 : SignFromVanishingOrder(divisor, x, dest, side)) is not { } divisorSign)
+            if ((IsNeverNegative(divisor, x) ? 1 : SignFromVanishingOrder(divisor, x, dest, side)) is not { } divisorSign)
                 return null;
             return dividendLimit.IsNegative == divisorSign > 0
                 ? Real.NegativeInfinity
@@ -1121,12 +1121,16 @@ internal static bool IsDeterminate(Entity combined, Variable x)
         /// from each side and NaN from both, where it is +oo. Where such an expression vanishes
         /// it does so from above, whatever its argument does.
         /// </remarks>
-        internal static bool IsNeverNegative(Entity expr) => expr switch
+        internal static bool IsNeverNegative(Entity expr, Variable x) => expr switch
         {
             Absf => true,
-            Powf(var @base, var exponent) => IsNeverNegative(@base) && exponent.Evaled is Real { IsNaN: false },
-            Mulf(var multiplier, var multiplicand) => IsNeverNegative(multiplier) && IsNeverNegative(multiplicand),
-            Divf(var dividend, var divisor) => IsNeverNegative(dividend) && IsNeverNegative(divisor),
+            // An even power of something real: x^2, and sinh(x)^2, whose powers to a negative
+            // exponent were NaN at 0 from both sides for want of it. The base has to be real for
+            // real x, since sqrt(x)^2 is x.
+            Powf(var @base, Integer { EInteger.IsEven: true }) when TreeAnalyzer.IsRealValued(@base, x) => true,
+            Powf(var @base, var exponent) => IsNeverNegative(@base, x) && exponent.Evaled is Real { IsNaN: false },
+            Mulf(var multiplier, var multiplicand) => IsNeverNegative(multiplier, x) && IsNeverNegative(multiplicand, x),
+            Divf(var dividend, var divisor) => IsNeverNegative(dividend, x) && IsNeverNegative(divisor, x),
             _ when expr.Evaled is Real { IsNegative: false, IsNaN: false } => !expr.FreeVariables.Any(),
             _ => false
         };

Sources/Tests/UnitTests/Calculus/AnEvenPowerVanishesFromAboveTest.cs

@@ -0,0 +1,45 @@
+//
+// 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.Core;
+using Xunit;
+
+namespace AngouriMath.Tests.Calculus
+{
+    /// <summary>
+    /// <c>1/((x^2)^tanh(1))</c> at 0 was NaN -- the claim that the limit does not exist -- where it is +oo from both sides.
+    /// A base was read as never negative only for an absolute value, or a power, product or quotient of never-negative
+    /// parts, so <c>x^2</c> was not, and a vanishing power of it to a constant that is not a number, or to an exponent
+    /// tending to a negative number, as <c>(sinh(x)^2)^(sin(x) - 1)</c>, was left unevaluated from each side. An even
+    /// power of a base real for real <c>x</c> is never negative now. The same on 2.6.0. Found by limitcheck at
+    /// <c>DEPTH=3</c>, <c>SEED=36</c>, and <c>sqrt(x^2)^tan(pi - 1)</c> at
+    /// <c>SEED=29</c>; no issue was filed.
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AnEvenPowerVanishesFromAboveTest
+    {
+        [Theory]
+        [InlineData("1/((x^2)^tanh(1))", "0", "+oo")]
+        [InlineData("-1/((x^2)^tanh(1))", "0", "-oo")]
+        [InlineData("(sinh(x)^2)^(sin(x) - 1)", "0", "+oo")]
+        [InlineData("((x - 1)^2)^(-pi/4)", "1", "+oo")]
+        [InlineData("(sin(x)^4)^(-tanh(1))", "0", "+oo")]
+        [InlineData("sqrt(x^2)^tan(pi - 1)", "0", "+oo")]
+        public void IsTheLimitFromEachSideAndBoth(string input, string destination, string expected)
+        {
+            foreach (var side in new[] { ApproachFrom.Left, ApproachFrom.Right, ApproachFrom.BothSides })
+                Assert.Equal(MathS.FromString(expected), MathS.FromString(input).Limit("x", destination, side));
+        }
+
+        // (i x)^2 is -x^2, negative on both sides, so a power of it to an exponent that is not whole is not real there,
+        // and is not claimed to diverge to +oo.
+        [Fact]
+        public void AnEvenPowerOfSomethingNotRealIsNotClaimed()
+            => Assert.NotEqual(MathS.FromString("+oo"),
+                MathS.FromString("1/(((i*x)^2)^tanh(1))").Limit("x", 0, ApproachFrom.Right));
+    }
+}