Home›Calculus 1›Limit laws with given limits›Problem 1.511

Limit laws with given limits

Problem 1.511 · easy

Suppose \( \displaystyle \lim_{x\to 3} f(x) = -5 \), \( \displaystyle \lim_{x\to 3} g(x) = 2 \) and \( \displaystyle \lim_{x\to 3} h(x) = -2 \). Find \( \displaystyle \lim_{x\to 3} \left(- f(x) + \left[h(x)\right]^{3}\right) \).

Each piece has a finite limit, and no denominator's limit is 0, so the limit laws (power and difference) let us replace every function by its limit.
Reviewed
\[ \left(-2\right)^{3} + 5 = -3 \]
Substitute the three limits and simplify.✓ Proved

Answer \( -3 \)

✓ 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 prose justification, not an algebraic equivalence. The limit laws (constant multiple, sum, power) do apply because every function has a finite limit, so substituting the limits is valid. The remark about denominators is vacuous but harmless.
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: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly substitutes +5 for -f(x) instead of -(-5). The limit of -f(x) is 5, so the expression should be 5 + (-8) = -3. Wait, let me

Senior review claude-sonnet-5-5, 2026-10-10: pass — The answer -3 is correct: -(-5) + (-2)^3 = 5 - 8 = -3. The mention of 'no denominator' is vacuous because the expression has no denominator, and the limit laws could be named as sum, constant multiple and power rather than 'difference'. Neither point is an error.

qwen3.6:27b-mlx: dismiss — The objection misreads line 2: it uses +5 for -f(x), which is -(-5) = 5 and is correct, and (-2)**3 + 5 = -3 holds.

Every verdict on record (5)

gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: inconclusive 2026-10-10 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly substitutes +5 for -f(x) instead of -(-5). The limit of -f(x) is 5, so the expression should be 5 + (-8) = -3. Wait, let me
claude-sonnet-5-5: pass 2026-10-10 — The answer -3 is correct: -(-5) + (-2)^3 = 5 - 8 = -3. The mention of 'no denominator' is vacuous because the expression has no denominator, and the limit laws could be named as sum, constant multiple and power rather than 'difference'. Neither point is an error.
qwen3.6:27b-mlx: inconclusive 2026-10-10 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly substitutes -5 for -f(x) instead of 5. Since lim f(x) = -5, lim -f(x) should be -(-5) = 5, but the equation line shows '(-2
gpt-oss:20b: pass 2026-10-10

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.