Limit of \( \displaystyle \frac{3 x + \left(3 x - 1\right)^{2} - 7}{3 x - 3} \) as \( x \to 1 \)
Problem 1.321 · medium
- \[ \lim_{x \to 1^+}\left(\frac{3 x + \left(3 x - 1\right)^{2} - 7}{3 x - 3}\right) \]limitStart with the initial limit expression.✓ Proved
- \[ = \lim_{x \to 1^+}\left(\frac{9 x^{2} - 3 x - 6}{3 x - 3}\right) \]simplify simplify factorExpand the squared term. Combine like terms in the numerator. Factor out the common constant 3.✓ Proved
- \[ = \lim_{x \to 1^+}\left(\frac{\left(x - 1\right) \left(9 x + 6\right)}{3 x - 3}\right) \]factorFactor the quadratic expression in the numerator.✓ Proved
- \[ = \lim_{x \to 1^+}\left(3 x + 2\right) \]cancel simplifyCancel the common factor (x - 1). Simplify the remaining expression.✓ Proved
- \[ = 5 \]limitEvaluate the limit at x = 1.✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. 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 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x - 3 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x - 3 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x - 3 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x - 3 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x - 3 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy took the limit from both sides and got the stated value |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — Step 6 incorrectly simplifies the denominator from 3*(x-1) to 3/3, which is algebraically invalid. The step should have resulted in (3*x + 2)/3 or similar, not (3*x + 2)/(3/3).
Senior review claude-sonnet-5-5, 2026-10-06: pass — Every line is SymPy-proved equivalent, the rule labels fit, and the answer 5 is correct (direct evaluation of 3x+2 at x=1). The unsimplified 3/3 in line 6 is a cosmetic quirk only.
gpt-oss:20b: dismiss — Line 6 is mathematically equal to line 5, since (3x+2)/(3/3) = 3x+2, and cancelling common factors is the single 'cancel' rule, so there is no false statement, wrong label, or independent second step; the awkward 3/3 form is cosmetic.
Every verdict on record (5)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — Step 6 incorrectly simplifies the denominator from 3*(x-1) to 3/3, which is algebraically invalid. The step should have resulted in (3*x + 2)/3 or similar, not (3*x + 2)/(3/3).claude-sonnet-5-5: pass 2026-10-06 — Every line is SymPy-proved equivalent, the rule labels fit, and the answer 5 is correct (direct evaluation of 3x+2 at x=1). The unsimplified 3/3 in line 6 is a cosmetic quirk only.qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly expands, factors, and cancels terms to evaluate the limit. Each step applies a single rule and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-10-06 — Step 6 applies two simplifications at once: it cancels the factor (x-1) and also divides the remaining 3 by 3, which should be a separate cancel step. The resulting expression should be (3*x+2), not (3*x+2)/(3/3).
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
gemma4:26b, checked 2026-10-06 with SymPy 1.14.0.