Limit laws with given limits
Problem 1.438 · easy
Suppose \( \displaystyle \lim_{n\to\infty} a_n = 2 \), \( \displaystyle \lim_{n\to\infty} b_n = 8 \) and \( \displaystyle \lim_{n\to\infty} c_n = -4 \). Find \( \displaystyle \lim_{n\to\infty} \left(\left(a_n - b_n\right)^{2}\right) \).
- Each piece has a finite limit, and no denominator's limit is 0, so the limit laws (power) let us replace every function by its limit.Reviewed
- \[ \left(-8 + 2\right)^{2} = 36 \]Substitute the three limits and simplify.✓ Proved
Answer \( 36 \)
✓ 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, not an equation. It correctly invokes the limit laws (difference and power) for finite limits. No denominators occur, so that clause holds vacuously. |
| 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: passqwen3.6:27b-mlx: fail (error) — The substitution in step 2 is incorrect: it calculates (-8 + 2)^2, which corresponds to (b_n - a_n)^2 or -(a_n - b_n)^2, but the problem asks for (a_n - b_n)^2. Since (2 - 8)^2 = (-6)^2 = 36 and (-8 + 2)^2 = (-6)^2 = 36, the final numerical answer happens to be correct, but the intermediate expression shown is algebraically inconsistent with the term (a_n - b_n) unless one assumes commutativity of subtraction inside the square without justification, or simply made a sign error that cancelled out. More importantly, the sentence in step 1 claims to replace 'every function' by its limit, but step 2 substitutes values in a way that suggests b_n came first (-8) then a_n (+2), implying the order was swapped. While the result is correct, the displayed substitution `(-8 + 2)` does not match `(a_n - b_n)` which should be `(2 - 8)`. This is misleading because it teaches incorrect substitution order.
Senior review claude-sonnet-5-5, 2026-10-08: pass — The substitution gives (2 - 8)^2 = (-8 + 2)^2 = 36, which is correct. The unused c_n and the vacuous 'no denominator' remark in line 1 are harmless.
qwen3.6:27b-mlx: dismiss — (-8 + 2) equals 2 - 8 = a_n - b_n in the limit, since -8 + 2 is just the terms reordered; b_n - a_n would be 8 - 2 = 6, so there is no order error, and SymPy proved the line.
Every verdict on record (5)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The substitution in step 2 is incorrect: it calculates (-8 + 2)^2, which corresponds to (b_n - a_n)^2 or -(a_n - b_n)^2, but the problem asks for (a_n - b_n)^2. Since (2 - 8)^2 = (-6)^2 = 36 and (-8 + 2)^2 = (-6)^2 = 36, the final numerical answer happens to be correct, but the intermediate expression shown is algebraically inconsistent with the term (a_n - b_n) unless one assumes commutativity of subtraction inside the square without justification, or simply made a sign error that cancelled out. More importantly, the sentence in step 1 claims to replace 'every function' by its limit, but step 2 substitutes values in a way that suggests b_n came first (-8) then a_n (+2), implying the order was swapped. While the result is correct, the displayed substitution `(-8 + 2)` does not match `(a_n - b_n)` which should be `(2 - 8)`. This is misleading because it teaches incorrect substitution order.claude-sonnet-5-5: pass 2026-10-08 — The substitution gives (2 - 8)^2 = (-8 + 2)^2 = 36, which is correct. The unused c_n and the vacuous 'no denominator' remark in line 1 are harmless.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The substitution in step 2 is incorrect: it calculates (-8 + 2)^2 = 36, which corresponds to (b_n - a_n)^2, but the problem asks for (a_n - b_n)^2, which is (2 - 8)^2 = 36. While the final numerical answer happens to be correct because the square eliminates the sign error, the intermediate step explicitly substitutes the values in the wrong order (b_n first, then a_n), modeling the expression as (b_n - a_n)^2 instead of (a_n - b_n)^2. This is a logical error in the application of the limit laws to the specific expression structure.
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 2026-10-08 with SymPy 1.14.0.