Home›Calculus 1›Limit laws with given limits›Problem 1.216

Limit laws with given limits

Problem 1.216 · easy

Suppose \( \displaystyle \lim_{x\to 5} f(x) = 6 \), \( \displaystyle \lim_{x\to 5} g(x) = 1 \) and \( \displaystyle \lim_{x\to 5} h(x) = -2 \). Find \( \displaystyle \lim_{x\to 5} \left(\left(f(x) - g(x)\right)^{2}\right) \).

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.
Reviewed
\[ \left(-1 + 6\right)^{2} = 25 \]
Substitute the three limits and simplify.✓ Proved

Answer \( 25 \)

✓ 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	claude-sonnet-5-5	claude-sonnet-5-5: This is a sentence. It invokes the limit laws (difference and power) for functions with finite limits. The denominator remark is vacuously true, since there are no denominators, and the laws do apply.
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: pass
qwen3.6:27b-mlx: fail (error) — The solution incorrectly substitutes -1 for the limit of g(x), which is given as 1. This leads to the wrong calculation (-1+6)^2=25 instead of the correct (1-6)^2=25 (wait, 1-6 is -5, squared is 25). Actually, let's re-read carefully. Limit of g is 1. Limit of f is 6. f-g = 6-1 = 5. 5^2 = 25. The solution says (-1+6)^2. Where did -1 come from? It seems to have used the limit of h(x) (-2) or just made a sign error on g(x) but got the right answer by coincidence? No, -1 is not any of the limits. Wait, if it used h(x)=-2, then (-2+6)^2 = 16. If it used g(x)=1, then (1-6)^2 = 25. The solution writes (-1+6)^2. This implies it substituted -1 for g(x). This is an error in substitution, even though the final numerical answer 25 happens to be correct for the expression (f-g)^2 if g=1. But the intermediate step is factually wrong regarding the value of g. A student would learn that g(x) approaches -1 or that one can arbitrarily change signs. This is a severe error in reasoning/substitution.

Senior review claude-sonnet-5-5, 2026-10-04: pass — The answer 25 is correct: the limit laws give (6-1)^2 = 25. Line 1's mention of denominators and line 2's 'three limits' (h is unused) are slightly loose but not false.

qwen3.6:27b-mlx: dismiss — The -1 in line 2 is the limit of -g(x), not a mis-substitution of g; (-1+6)^2 = (6-1)^2 = 25, so the answer is correct, and the reviewer's own text concedes (6-1)^2 = 25.

Every verdict on record (5)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution incorrectly substitutes -1 for the limit of g(x), which is given as 1. This leads to the wrong calculation (-1+6)^2=25 instead of the correct (1-6)^2=25 (wait, 1-6 is -5, squared is 25). Actually, let's re-read carefully. Limit of g is 1. Limit of f is 6. f-g = 6-1 = 5. 5^2 = 25. The solution says (-1+6)^2. Where did -1 come from? It seems to have used the limit of h(x) (-2) or just made a sign error on g(x) but got the right answer by coincidence? No, -1 is not any of the limits. Wait, if it used h(x)=-2, then (-2+6)^2 = 16. If it used g(x)=1, then (1-6)^2 = 25. The solution writes (-1+6)^2. This implies it substituted -1 for g(x). This is an error in substitution, even though the final numerical answer 25 happens to be correct for the expression (f-g)^2 if g=1. But the intermediate step is factually wrong regarding the value of g. A student would learn that g(x) approaches -1 or that one can arbitrarily change signs. This is a severe error in reasoning/substitution.
gpt-oss:20b: pass 2026-10-04
claude-sonnet-5-5: pass 2026-10-04 — The answer 25 is correct: the limit laws give (6-1)^2 = 25. Line 1's mention of denominators and line 2's 'three limits' (h is unused) are slightly loose but not false.
qwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly substitutes g(x) as -1 instead of 1, leading to the wrong answer. The correct calculation is (6 - 1)^2 = 25, but the text i
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.