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Limit laws with given limits

Problem 1.781 · easy

Suppose \( \displaystyle \lim_{x\to 0} f(x) = -3 \), \( \displaystyle \lim_{x\to 0} g(x) = 4 \) and \( \displaystyle \lim_{x\to 0} h(x) = 6 \). Find \( \displaystyle \lim_{x\to 0} \left(\left(f(x) - g(x)\right)^{2}\right) \).

Each piece has a finite limit, and no denominator's limit is 0, so the limit laws (power) let us replace every function by its limit.
\[ \left(-4 - 3\right)^{2} = 49 \]
Substitute the three limits and simplify.✓ Proved

Answer \( 49 \)

Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the limit of the same combination of concrete functions with those limits, taken by SymPy

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The substitution in step 2 is incorrect; it calculates (-4 - 3)^2 instead of the correct (-3 - 4)^2. While the final answer 49 is coincidentally correct because (-7)^2 = 49, the intermediate arithmetic shown is wrong, which would teach a student incorrect substitution procedures.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-31 — The substitution in step 2 is incorrect; it calculates (-4 - 3)^2 instead of the correct (-3 - 4)^2. While the final answer 49 is coincidentally correct because (-7)^2 = 49, the intermediate arithmetic shown is wrong, which would teach a student incorrect substitution procedures.
gpt-oss:20b: pass 2026-10-31
qwen3.6:27b-mlx: fail (error) 2026-10-30 — The substitution in step 2 is incorrect: it computes (-4 - 3)^2, implying f(x) -> 4 and g(x) -> 3, whereas the problem states f(x) -> -3 and g(x) -> 4. The correct calculation should be (-3 - 4)^2 = 49, but the written steps show the wrong values being substituted.
gpt-oss:20b: pass 2026-10-30
qwen3.6:27b-mlx: fail (error) 2026-10-30 — The solution incorrectly substitutes the limit of f(x) as -4 instead of the given value -3. The correct calculation should be (-3 - 4)^2 = 49, but the text explicitly states '(-4 - 3)**2', implying f(x) -> -4.
gpt-oss:20b: pass 2026-10-30

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/limit_laws_values, checked 2026-10-31 with SymPy 1.14.0.