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Limit laws with given limits

Problem 1.509 · easy

Suppose \( \displaystyle \lim_{n\to\infty} a_n = 6 \), \( \displaystyle \lim_{n\to\infty} b_n = 7 \) and \( \displaystyle \lim_{n\to\infty} c_n = -4 \). 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
\[ - \frac{7}{-4} + 6 \cdot 4 = \frac{103}{4} \]
Substitute the three limits and simplify.✓ Proved

Answer \( \frac{103}{4} \)

✓ 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	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly applies limit laws for arithmetic combinations of convergent sequences. The algebraic substitution and simplification are correct.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies limit laws for arithmetic combinations of convergent sequences. The algebraic substitution and simplification are correct.
gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies the limit laws for sums, products, and quotients, as the limit of the denominator is non-zero. The arithmetic is correct.

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 2026-10-10 with SymPy 1.14.0.