Turn l'Hôpital away from a quotient whose whole powers grow

BREAKING-CHANGES.md

@@ -351,6 +351,20 @@ positive on both sides. `x^x` is still `NaN` at 0.
 | `"(x^2)^x".ToEntity().Limit("x", 0)` | `NaN` | `1` |
 | `"(x*x)^(x - 1)".ToEntity().Limit("x", 0)` | `NaN` | `+oo` |
 
+### A 0^0 limit whose l'Hopital chain doubles its powers has its value
+
+**Silent: a different answer.** l'Hopital's rule was turned away where a
+quotient grew in nodes, and a chain whose divisor is a reciprocal power, `x^n / (1 / g^k)`, grows in its exponents
+instead: the derivative of `1/g^k` is over `g^(2k)`, so k doubled at each step and the chain never ended. The whole
+exponents are measured as the nodes are now, with the same room, and the limits below have their value. No issue was
+filed; a check of limits against values near the destination found it.
+
+| Input | Was (2.6.0) | Now |
+|---|---|---|
+| `limitright((log(x, 1/2))^(exp(x) - sgn(x)), x, 0)` | `0` | `1` |
+
+The same spelled `(-ln(2)/ln(x))^(exp(x) - 1)` was `1` in 2.6.0 and did not return after it; it is `1` again.
+
 ### A limit of a logarithm that is not decided is left unevaluated
 
 **Not silent: what overflowed the stack now returns.** A limit of a logarithm whose argument has no limit, as

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

@@ -1264,7 +1264,23 @@ private static bool AlreadyBeingDifferentiated(Entity quotient)
         /// unchanged at 112/117 and, in total time, within noise of where it was.
         /// </remarks>
         private static bool GrewTooMuch(Entity quotient, Entity applied)
-            => applied.Nodes.Count() > Math.Max(quotient.Nodes.Count() + 8, quotient.Nodes.Count() * 8 / 5);
+            => applied.Nodes.Count() > Math.Max(quotient.Nodes.Count() + 8, quotient.Nodes.Count() * 8 / 5)
+               || WholePowers(applied) > Math.Max(WholePowers(quotient) + 8, WholePowers(quotient) * 8 / 5);
+
+        /// <summary>
+        /// The whole exponents of <paramref name="expr"/>'s powers, added up: the growth a node count does not see.
+        /// </summary>
+        /// <remarks>
+        /// <c>(e^(-1/x) - 1)^12</c> and <c>(e^(-1/x) - 1)^24</c> have the same nodes, and a chain whose divisor is
+        /// a reciprocal power, <c>x^n / (1 / g^k)</c>, doubles k at every step, since the derivative of
+        /// <c>1/g^k</c> is over <c>g^(2k)</c>: the limit of <c>(-ln(2)/ln(x))^(e^x - 1)</c> at 0 from the right
+        /// went 6, 12, 24, ... and did not finish in ten minutes, the inner limits expanding the powers. A
+        /// derivative lowers the degrees of a chain that is getting somewhere; so the same room as for the
+        /// nodes, the larger of eight more and three fifths again.
+        /// </remarks>
+        private static int WholePowers(Entity expr)
+            => expr.Nodes.Sum(static node => node is Powf(_, Integer { EInteger: var exponent }) && exponent.CanFitInInt32()
+                ? Math.Abs(exponent.ToInt32Unchecked()) : 0);
 
         [ThreadStatic] private static bool suppresslHopital;
 

Sources/Tests/UnitTests/Calculus/AQuotientWhosePowersGrowIsNotDifferentiatedTest.cs

@@ -0,0 +1,46 @@
+//
+// 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 System;
+using System.Threading.Tasks;
+using AngouriMath.Core;
+using Xunit;
+using static AngouriMath.Entity;
+
+namespace AngouriMath.Tests.Calculus
+{
+    /// <summary>
+    /// l'Hopital's rule was turned away where a quotient grew in nodes, and a chain whose divisor is a reciprocal power,
+    /// <c>x^n / (1 / g^k)</c>, grows in its exponents instead: the derivative of <c>1/g^k</c> is over <c>g^(2k)</c>, so k
+    /// went 6, 12, 24, ..., and the limit of <c>(-ln(2)/ln(x))^(e^x - 1)</c> at 0 from the right did not finish in ten
+    /// minutes. It is 1, as 2.6.0 answered; the spelling with <c>log(x, 1/2)</c> and <c>sgn(x)</c>, 2.6.0 answered 0. Found by
+    /// limitcheck at <c>DEPTH=3</c>, <c>SEED=23</c>; no issue was filed.
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AQuotientWhosePowersGrowIsNotDifferentiatedTest
+    {
+        [Theory]
+        [InlineData("(-ln(2)/ln(x))^(exp(x) - 1)")]
+        [InlineData("(log(x, 1/2))^(exp(x) - sgn(x))")]
+        public void IsOneAtZeroFromTheRight(string input)
+        {
+            var task = Task.Factory.StartNew(() => MathS.FromString(input).Limit("x", 0, ApproachFrom.Right),
+                TaskCreationOptions.LongRunning);
+            // Far from both: the limit takes about 6 s here and past a minute on CI's Windows runner, whose suite runs
+            // three times as long, and before the fix it did not finish in ten minutes.
+            Assert.True(task.Wait(TimeSpan.FromMinutes(5)), $"the limit of {input} at 0 from the right did not finish in five minutes");
+            Assert.Equal(Number.Integer.One, task.Result);
+        }
+
+        [Theory]
+        [InlineData("1/x^2 - 1/sin(x)^2", "-1/3")]
+        [InlineData("(1 - cos(x)) / x^2", "1/2")]
+        [InlineData("x^3 / (x - sin(x))", "6")]
+        public void AChainThatGetsSomewhereIsAsBefore(string input, string expected)
+            => Assert.Equal(MathS.FromString(expected), MathS.FromString(input).Limit("x", 0));
+    }
+}