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Limit laws with given limits

Problem 1.508 · easy

Suppose $ \displaystyle \lim_{n\to\infty} a_n = -1 $, $ \displaystyle \lim_{n\to\infty} b_n = 3 $ and $ \displaystyle \lim_{n\to\infty} c_n = 3 $. Find $ \displaystyle \lim_{n\to\infty} \left(\sqrt{\left[a_n\right]^{2} + \left[b_n\right]^{2}}\right) $.

Each piece has a finite limit, and no denominator's limit is 0, so the limit laws (root) let us replace every function by its limit.
Reviewed
\[ \sqrt{\left(-1\right) \left(-1\right) + 3 \cdot 3} = \sqrt{10} \]
Substitute the three limits and simplify.✓ Proved

Answer $ \sqrt{10} $

✓ 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 sentence justifying the limit laws, not an equation. The brackets are grouping, the expression is a continuous function of a_n and b_n, and the mention of denominators is vacuous but harmless.
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 continuous functions (square root and polynomial operations) to the given limits.

Senior review claude-sonnet-5-5, 2026-10-10: pass — The limit is sqrt((-1)^2+3^2)=sqrt(10), which matches the stated answer. The remarks about a denominator and 'the three limits' are slightly loose, since c_n does not appear and there is no denominator, but nothing in the solution is false.

qwen3.6:27b-mlx: dismiss — The square brackets in [a_n]^2 are ordinary grouping brackets, not a floor function, so the sequences are continuous in the expression and the limit laws apply directly.

Every verdict on record (5)

gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies the limit laws for continuous functions (square root and polynomial operations) to the given limits.
claude-sonnet-5-5: pass 2026-10-10 — The limit is sqrt((-1)^2+3^2)=sqrt(10), which matches the stated answer. The remarks about a denominator and 'the three limits' are slightly loose, since c_n does not appear and there is no denominator, but nothing in the solution is false.
gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution incorrectly applies limit laws to the floor function, which is discontinuous. Since $a_n \to -1$, $[a_n]$ does not necessarily converge to $[-1] = -1$ (e.g., if $a_n = -1 + 1/n$, $[a_n] = -2$). The limit depends on the direction of approach, so it is not uniquely determined by the limits of $a_n, b_n, c_n$ alone.

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.