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Limit laws with given limits

Problem 1.569 · easy

Suppose \( \displaystyle \lim_{n\to\infty} a_n = -4 \), \( \displaystyle \lim_{n\to\infty} b_n = 3 \) and \( \displaystyle \lim_{n\to\infty} c_n = -4 \). Find \( \displaystyle \lim_{n\to\infty} \left(a_n b_n - c_n\right) \).

Each piece has a finite limit, and no denominator's limit is 0, so the limit laws (product and difference) let us replace every function by its limit.
Reviewed
\[ 4 - 4 \cdot 3 = -8 \]
Substitute the three limits and simplify.✓ Proved

Answer \( -8 \)

Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see the receipt.

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	claude-sonnet-5-5	claude-sonnet-5-5: This sentence only invokes the product and difference limit laws, which apply because all three limits are finite. The mention of denominators is superfluous but harmless. It is not an algebraic equivalence, so no script applies.
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 solution incorrectly substitutes 4 for the limit of a_n, which is given as -4. The correct calculation should be (-4)(3) - (-4) = -8, but the reasoning shown (4 - 12) is factually wrong regarding the input values, even though the final numerical answer happens to be correct by coincidence.

Senior review claude-sonnet-5-5, 2026-10-12: fail — The final answer -8 is correct, but line 2 is labeled as a substitution of the limits and does not display that substitution (it should read (-4)*3 - (-4) = -12 + 4 = -8). A student would learn a wrong substitution. The remark about denominators in line 1 is irrelevant, since the expression has none, but it is not false.

qwen3.6:27b-mlx: uphold — Substituting the limits gives (-4)(3) - (-4), not 4 - 4*3; the displayed line drops the signs and only equals -8 by coincidence, so it does not show the substitution it claims to.

Every verdict on record (5)

gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution incorrectly substitutes 4 for the limit of a_n, which is given as -4. The correct calculation should be (-4)(3) - (-4) = -8, but the reasoning shown (4 - 12) is factually wrong regarding the input values, even though the final numerical answer happens to be correct by coincidence.
claude-sonnet-5-5: fail (misleading) 2026-10-12 — The final answer -8 is correct, but line 2 is labeled as a substitution of the limits and does not display that substitution (it should read (-4)*3 - (-4) = -12 + 4 = -8). A student would learn a wrong substitution. The remark about denominators in line 1 is irrelevant, since the expression has none, but it is not false.
qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution substitutes the wrong values into the expression. It calculates 4 - 4*3, but the problem states lim a_n = -4 and lim c_n = -4. The correct substitution should be (-4)*3 - (-4) = -12 + 4 = -8. Although the final answer happens to be correct due to a coincidence of signs, the intermediate step shown is mathematically incorrect and misleading.
gpt-oss:20b: pass 2026-10-12

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 2026-10-12 with SymPy 1.14.0.