Limit laws with given limits
Problem 1.436 · easy
Suppose \( \displaystyle \lim_{x\to 3} f(x) = 8 \), \( \displaystyle \lim_{x\to 3} g(x) = 8 \) and \( \displaystyle \lim_{x\to 3} h(x) = -5 \). Find \( \displaystyle \lim_{x\to 3} \left(\left(f(x) - g(x)\right)^{2}\right) \).
- 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.
- \[ \left(-8 + 8\right)^{2} = 0 \]Substitute the three limits and simplify.✓ Proved
Answer \( 0 \)
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 -8 for the limit of f(x), which is given as 8. This arithmetic error leads to the wrong intermediate value, although the final answer happens to be correct by coincidence.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution incorrectly substitutes -8 for the limit of f(x), which is given as 8. This arithmetic error leads to the wrong intermediate value, although the final answer happens to be correct by coincidence.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies the limit laws for subtraction and powers, as the individual limits exist and are finite. 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 2026-10-08 with SymPy 1.14.0.