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Limit laws with given limits

Problem 1.565 · easy

Suppose \( \displaystyle \lim_{n\to\infty} a_n = -4 \), \( \displaystyle \lim_{n\to\infty} b_n = -4 \) and \( \displaystyle \lim_{n\to\infty} c_n = -5 \). Find \( \displaystyle \lim_{n\to\infty} \left(2 a_n + 3 b_n\right) \).

Each piece has a finite limit, and no denominator's limit is 0, so the limit laws (sum and constant multiple) let us replace every function by its limit.
Reviewed
\[ \left(-4\right) 3 - 4 \cdot 2 = -20 \]
Substitute the three limits and simplify.✓ Proved

Answer \( -20 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	claude-sonnet-5-5	claude-sonnet-5-5: This is a sentence, not an algebraic equivalence. The sum and constant-multiple limit laws apply to 2a_n + 3b_n because the limits are finite. There are no denominators, so the condition about them holds vacuously. The limit of c_n is not needed.
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 references 'no denominator's limit is 0', which is irrelevant for a linear combination of sequences. Additionally, the substitution in the equation is inconsistent with the text, using -4 for both a_n and b_n but structuring the calculation as if c_n were involved or simply mislabeling the terms.

Senior review claude-sonnet-5-5, 2026-10-12: pass — Both limits are -4, so 2(-4)+3(-4) = -20, and the line's pairing of coefficients is numerically harmless. The sentence's mention of denominators is unneeded boilerplate but not false, and c_n is simply not used.

qwen3.6:27b-mlx: dismiss — The remark about denominators is vacuously true and not a false statement. Line 2 is marked 'proved', not 'unchecked', and -4*3 - 4*2 = -20 holds, so there is no real defect.

Every verdict on record (5)

gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution incorrectly references 'no denominator's limit is 0', which is irrelevant for a linear combination of sequences. Additionally, the substitution in the equation is inconsistent with the text, using -4 for both a_n and b_n but structuring the calculation as if c_n were involved or simply mislabeling the terms.
claude-sonnet-5-5: pass 2026-10-12 — Both limits are -4, so 2(-4)+3(-4) = -20, and the line's pairing of coefficients is numerically harmless. The sentence's mention of denominators is unneeded boilerplate but not false, and c_n is simply not used.
gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution incorrectly references a denominator and the limit of c_n, which are irrelevant to the expression 2a_n + 3b_n. Additionally, the substitution step is labeled 'unchecked' despite being a simple arithmetic calculation that should be verified.

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.