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Limit laws with given limits

Problem 1.776 · easy

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

Each piece has a finite limit, and no denominator's limit is 0, so the limit laws (power and difference) let us replace every function by its limit.
\[ 3 + 5^{3} = 128 \]
Substitute the three limits and simplify.✓ Proved

Answer \( 128 \)

Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Not checked	—	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: fail (error) — The solution incorrectly evaluates the limit of the floor function [c_n] as 5, ignoring that for c_n approaching 5 from below (which is possible), [c_n] would be 4. The limit of the floor function does not necessarily equal the floor of the limit.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-31 — The solution incorrectly evaluates the limit of the floor function [c_n] as 5, ignoring that for c_n approaching 5 from below (which is possible), [c_n] would be 4. The limit of the floor function does not necessarily equal the floor of the limit.
gpt-oss:20b: pass 2026-10-31
qwen3.6:27b-mlx: fail (style) 2026-10-30 — [domain objection, downgraded to style] The solution incorrectly applies the limit law for powers to the floor function, assuming lim [c_n]^3 = [lim c_n]^3. Since c_n approaches 5 from an unknown direction, the floor of c_n could be 4 or 5, making the limit undefined or dependent on the sequence's approach, not necessarily 125.
gpt-oss:20b: pass 2026-10-30
qwen3.6:27b-mlx: inconclusive 2026-10-30 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly substitutes -3 for -a_n, resulting in +3, whereas the limit of -a_n is 3. However, the calculation 3 + 125 = 128 is arithme
gpt-oss:20b: pass 2026-10-30

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 2026-10-31 with SymPy 1.14.0.