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Limit laws with given limits

Problem 1.210 · easy

Suppose \( \displaystyle \lim_{n\to\infty} a_n = 9 \), \( \displaystyle \lim_{n\to\infty} b_n = 4 \) and \( \displaystyle \lim_{n\to\infty} c_n = 4 \). 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
\[ - \frac{4}{4} + 9 \cdot 4 = 35 \]
Substitute the three limits and simplify.✓ Proved

Answer \( 35 \)

✓ 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 sums, products, and quotients, noting that the denominator's limit is non-zero. The arithmetic is correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies limit laws for sums, products, and quotients, noting that the denominator's limit is non-zero. The arithmetic is correct.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies limit laws for arithmetic combinations of sequences. The substitution and arithmetic are correct.
gpt-oss:20b: pass 2026-10-04

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 2026-10-04 with SymPy 1.14.0.