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Limit laws with given limits

Problem 1.566 · easy

Suppose $ \displaystyle \lim_{n\to\infty} a_n = -4 $, $ \displaystyle \lim_{n\to\infty} b_n = -5 $ and $ \displaystyle \lim_{n\to\infty} c_n = 6 $. 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.
\[ \sqrt{\left(-4\right) \left(-4\right) - 5 \left(-5\right)} = \sqrt{41} \]
Substitute the three limits and simplify.✓ Proved

Answer $ \sqrt{41} $

Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see the receipt.

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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The second line incorrectly computes the sum inside the square root: it uses $-4*(-4) - 5*(-5)$ instead of $(-4)^2 + (-5)^2$, leading to a wrong value
qwen3.6:27b-mlx: fail (error) — The solution incorrectly substitutes the limit of $c_n$ into the expression, which does not contain $c_n$. Additionally, the algebraic simplification in the equation line is incorrect: $-4^2 + (-5)^2 = 16 + 25 = 41$, but the written expression $-4*(-4) - 5*(-5)$ evaluates to $16 + 25 = 41$ only by coincidence of signs if interpreted as $a_n^2 + b_n^2$, but the notation is confusing and the inclusion of the irrelevant $c_n$ limit in the reasoning is a logical error.

Every verdict on record (4)

gpt-oss:20b: inconclusive 2026-10-12 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The second line incorrectly computes the sum inside the square root: it uses $-4*(-4) - 5*(-5)$ instead of $(-4)^2 + (-5)^2$, leading to a wrong value
qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution incorrectly substitutes the limit of $c_n$ into the expression, which does not contain $c_n$. Additionally, the algebraic simplification in the equation line is incorrect: $-4^2 + (-5)^2 = 16 + 25 = 41$, but the written expression $-4*(-4) - 5*(-5)$ evaluates to $16 + 25 = 41$ only by coincidence of signs if interpreted as $a_n^2 + b_n^2$, but the notation is confusing and the inclusion of the irrelevant $c_n$ limit in the reasoning is a logical error.
gpt-oss:20b: fail (error) 2026-10-12 — The solution incorrectly references a denominator that does not exist, ignores the sequence c_n entirely, and misstates the justification for applying the limit law. The setup is incomplete and misleading.
qwen3.6:27b-mlx: inconclusive 2026-10-12 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly applies limit laws to the floor function, which is discontinuous at integers. Since the limits of $a_n$ and $b_n$ are integ

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.