Limit laws with given limits
Problem 1.432 · easy
Suppose \( \displaystyle \lim_{x\to -4} f(x) = 8 \), \( \displaystyle \lim_{x\to -4} g(x) = -3 \) and \( \displaystyle \lim_{x\to -4} h(x) = -5 \). Find \( \displaystyle \lim_{x\to -4} \left(f(x) g(x) - h(x)\right) \).
- 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.
- \[ 8 \left(-3\right) + 5 = -19 \]Substitute the three limits and simplify.✓ Proved
Answer \( -19 \)
Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in 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: passqwen3.6:27b-mlx: fail (error) — The solution incorrectly substitutes +5 for h(x) instead of -5, leading to the wrong final answer of -19 instead of -29.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution incorrectly substitutes +5 for h(x) instead of -5, leading to the wrong final answer of -19 instead of -29.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies limit laws for products and differences. The arithmetic in the equation line (8*(-3) - (-5) = -24 + 5 = -19) is correct, despite the comment saying '+ 5' which is consistent with subtracting -5.
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.