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Limit laws with given limits

Problem 1.568 · easy

Suppose $ \displaystyle \lim_{n\to\infty} a_n = 7 $, $ \displaystyle \lim_{n\to\infty} b_n = -6 $ and $ \displaystyle \lim_{n\to\infty} c_n = 6 $. Find $ \displaystyle \lim_{n\to\infty} \left(4 a_n - \frac{b_n}{c_n}\right) $.

Each piece has a finite limit, and no denominator's limit is 0, so the limit laws (difference and quotient) let us replace every function by its limit.
Reviewed
\[ \frac{6}{6} + 7 \cdot 4 = 29 \]
Substitute the three limits and simplify.✓ Proved

Answer $ 29 $

✓ 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 prose justification, not an algebraic equivalence. The limits 7, -6 and 6 are all finite, and the denominator's limit 6 is nonzero, so the sum/difference, constant-multiple and quotient limit laws apply.
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 limit laws for sums, differences, products, and quotients, as the denominator's limit is non-zero. The arithmetic is correct.

Senior review claude-sonnet-5-5, 2026-10-12: pass — The limit laws apply because every limit is finite and lim c_n = 6 ≠ 0. The value is 4·7 − (−6)/6 = 29, which matches line 2 and the stated answer. Line 2 folds the substitution, the sign simplification and the reordering into one step, which is only a minor stylistic compression.

qwen3.6:27b-mlx: dismiss — The objection claims the solution uses 4×6, but line 2 uses 7*4 (the limit of a_n is 7). 4·7 − (−6)/6 = 28 + 1 = 29, so the arithmetic is correct.

Every verdict on record (5)

gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly applies limit laws for sums, differences, products, and quotients, as the denominator's limit is non-zero. The arithmetic is correct.
claude-sonnet-5-5: pass 2026-10-12 — The limit laws apply because every limit is finite and lim c_n = 6 ≠ 0. The value is 4·7 − (−6)/6 = 29, which matches line 2 and the stated answer. Line 2 folds the substitution, the sign simplification and the reordering into one step, which is only a minor stylistic compression.
qwen3.6:27b-mlx: inconclusive 2026-10-12 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly substitutes the limit of $a_n$ (which is 7) as 6 in the term $4a_n$, calculating $4 \\times 6$ instead of $4 \\times 7$. Th
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.