Limit laws with given limits
Problem 1.775 · easy
Suppose \( \displaystyle \lim_{x\to -2} f(x) = 1 \), \( \displaystyle \lim_{x\to -2} g(x) = 7 \) and \( \displaystyle \lim_{x\to -2} h(x) = -1 \). Find \( \displaystyle \lim_{x\to -2} \left(\left(f(x) - g(x)\right)^{2}\right) \).
- 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.
- \[ \left(-7 + 1\right)^{2} = 36 \]Substitute the three limits and simplify.✓ Proved
Answer \( 36 \)
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. How we verify
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 substitution in the equation is incorrect: it computes (-7 + 1)^2, which corresponds to (g(x) - f(x))^2 or -(f(x)-g(x))^2, but the term inside the square is f(x) - g(x) = 1 - 7 = -6. While (-6)^2 also equals 36, the intermediate step shown is algebraically inconsistent with the expression (f(x) - g(x))^2.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-31 — The substitution in the equation is incorrect: it computes (-7 + 1)^2, which corresponds to (g(x) - f(x))^2 or -(f(x)-g(x))^2, but the term inside the square is f(x) - g(x) = 1 - 7 = -6. While (-6)^2 also equals 36, the intermediate step shown is algebraically inconsistent with the expression (f(x) - g(x))^2.gpt-oss:20b: pass 2026-10-31qwen3.6:27b-mlx: fail (error) 2026-10-30 — The substitution in the equation is incorrect; it computes (1 - 7)^2 = 36, but the text implies substituting f, g, h directly into an expression that doesn't match the problem structure (f-g)^2. Specifically, the equation shows -7 + 1, which corresponds to g + f or similar, not f - g (which would be 1 - 7 = -6). While the final answer 36 is correct for (1-7)^2, the intermediate step shown (-7+1)**2 is algebraically inconsistent with the standard order of f-g unless f=-7 and g=1, which contradicts the given limits. Wait, 1-7 is -6. (-6)^2 is 36. The equation shows (-7+1)**2. -7+1 is -6. So the arithmetic is correct for the value, but the mapping of limits to variables is confusing/wrong in the text description vs the equation. The text says 'Substitute the three limits', but h is not used. The equation uses -7 and 1. g is 7, f is 1. So it used -g + f. This is f - g. So the math is actually correct: f=1, g=7. f-g = 1-7 = -6. The equation writes -7+1. This is -g+f. This is equal to f-g. So the calculation is correct. However, the sentence says 'replace every function by its limit'. It does not explicitly state the limit law for difference and power. It is a bit loose but not strictly wrong. Let's look closer. The sentence says 'no denominator's limit is 0'. There is no denominator. This is irrelevant but not false. The main issue is clarity. Is it an error? No. Is it misleading? Maybe slightly. But the prompt asks for adversarial review. The equation line is marked 'proved' by CAS. The sentence justifies it. The justification is weak ('no denominator...') but the conclusion is correct. Actually, looking at the equation: `(-7 + 1)**2`. This is `(-6)**2 = 36`. The limits are f=1, g=7. f-g = 1-7 = -6. So the substitution is correct. The sentence is just poorly phrased. I will mark it as style or pass? The prompt says 'find what is wrong'. The sentence claims 'replace every function by its limit'. It doesn't mention h, which is fine since h isn't in the expression. The 'no denominator' part is weird but not false. I'll lean towards pass or style. But wait, is there a subtle error? No. The answer is correct. The steps are correct. The explanation is just brief. I will give it a pass.gpt-oss:20b: pass 2026-10-30qwen3.6:27b-mlx: fail (error) 2026-10-30 — The substitution in step 2 is incorrect: it calculates (-7 + 1)^2, which corresponds to (g(-2) + h(-2))^2 or similar, instead of (f(-2) - g(-2))^2 = (1 - 7)^2 = 36. While the final numerical answer is coincidentally correct, the intermediate expression shown is wrong and misleading.gpt-oss:20b: pass 2026-10-30
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 2026-10-31 with SymPy 1.14.0.