Limit laws with given limits
Problem 1.213 · easy
Suppose \( \displaystyle \lim_{n\to\infty} a_n = -3 \), \( \displaystyle \lim_{n\to\infty} b_n = 2 \) and \( \displaystyle \lim_{n\to\infty} c_n = 6 \). Find \( \displaystyle \lim_{n\to\infty} \left(a_n b_n - c_n\right) \).
- Each piece has a finite limit, and no denominator's limit is 0, so the limit laws (product and difference) let us replace every function by its limit.Reviewed
- \[ \left(-3\right) 2 - 6 = -12 \]Substitute the three limits and simplify.✓ Proved
Answer \( -12 \)
✓ 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: passqwen3.6:27b-mlx: pass — The solution correctly applies the limit laws for products and differences, as the individual limits are finite. The calculation is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the limit laws for products and differences, as the individual limits are finite. The calculation is correct.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the limit laws for products and differences, as the individual limits are finite. The arithmetic is 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.