Limit laws with given limits
Problem 1.505 · easy
Suppose \( \displaystyle \lim_{x\to 3} f(x) = -4 \), \( \displaystyle \lim_{x\to 3} g(x) = 8 \) and \( \displaystyle \lim_{x\to 3} h(x) = -4 \). Find \( \displaystyle \lim_{x\to 3} \left(4 f(x) - \frac{g(x)}{h(x)}\right) \).
- Each piece has a finite limit, and no denominator's limit is 0, so the limit laws (difference and quotient) let us replace every function by its limit.Reviewed
- \[ \left(-4\right) 4 - \frac{8}{-4} = -14 \]Substitute the three limits and simplify.✓ Proved
Answer \( -14 \)
✓ 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 identifies that the limit laws apply because the individual limits exist and the denominator's limit is non-zero. The calculation is correct.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies that the limit laws apply because the individual limits exist and the denominator's limit is non-zero. The calculation is correct.gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies that the limit laws for differences and quotients apply because the individual limits exist and the denominator's limit is non-zero. The arithmetic is correct.
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.