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Limit laws with given limits

Problem 1.342 · easy

Suppose \( \displaystyle \lim_{n\to\infty} a_n = 9 \), \( \displaystyle \lim_{n\to\infty} b_n = -4 \) and \( \displaystyle \lim_{n\to\infty} c_n = 1 \). 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
\[ \left(-4\right) 3 + 9 \cdot 2 = 6 \]
Substitute the three limits and simplify.✓ Proved

Answer \( 6 \)

✓ 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 justification sentence, not an equation. The sum and constant-multiple limit laws apply because a_n and b_n have finite limits, and there is no denominator, so that clause holds vacuously. c_n is simply unused.
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: pass — The solution correctly applies the limit laws for sums and constant multiples, and the arithmetic is correct.

Senior review claude-sonnet-5-5, 2026-10-06: pass — The answer 6 is correct and the limit-law justification is sound (the denominator remark is vacuously true). It would read slightly better to say 'substitute the limits of a_n and b_n' and to show 2·9 + 3·(−4) before simplifying.

qwen3.6:27b-mlx: dismiss — The c_n limit is just a distractor given in the problem, and the equation correctly uses only the two relevant limits (2·9 + 3·(−4) = 6); the 'three limits' wording is loose but teaches nothing false.

Every verdict on record (5)

gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the limit laws for sums and constant multiples, and the arithmetic is correct.
claude-sonnet-5-5: pass 2026-10-06 — The answer 6 is correct and the limit-law justification is sound (the denominator remark is vacuously true). It would read slightly better to say 'substitute the limits of a_n and b_n' and to show 2·9 + 3·(−4) before simplifying.
gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: fail (misleading) 2026-10-06 — The solution mentions c_n in the text ('Substitute the three limits') but c_n is not part of the expression being evaluated. This is confusing and implies c_n is relevant to the calculation.

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 2026-10-06 with SymPy 1.14.0.