Limit of \( \displaystyle \frac{- 3 x + \left(3 x - 1\right)^{2} + 1}{\left(3 x - 1\right)^{2} - 1} \) as \( x \to \frac{2}{3} \)
Problem 1.221 · hard
Evaluate \( \displaystyle \lim_{x \to \frac{2}{3}} \frac{- 3 x + \left(3 x - 1\right)^{2} + 1}{\left(3 x - 1\right)^{2} - 1} \).
- \[ \lim_{x \to \frac{2}{3}^+}\left(\frac{- 3 x + \left(3 x - 1\right)^{2} + 1}{\left(3 x - 1\right)^{2} - 1}\right) \]limitStarting with the original limit expression.✓ Proved
- \[ = \lim_{x \to \frac{2}{3}^+}\left(\frac{9 x^{2} - 9 x + 2}{9 x^{2} - 6 x}\right) \]simplifyExpand the squared terms and simplify the numerator and denominator.✓ Proved
- \[ = \lim_{x \to \frac{2}{3}^+}\left(\frac{3 x - 1}{3 x}\right) \]factor cancelFactor both the numerator and the denominator. Cancel the common factor (3*x - 2).✓ Proved
- \[ = \frac{1}{2} \]limit simplifyEvaluate the remaining expression at x = 2/3. Simplify the resulting fraction.✓ Proved
- \[ = 0.5 \]limitFinal value of the limit.✓ Proved
Answer \( \frac{1}{2} \)
✓ 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 - 1)**2 - 1 = 0 undefined where 9*x**2 - 6*x = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 - 6*x = 0 undefined where x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 7 | ✓ 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: fail (style) — Step 5 applies a substitution (plugging in x=2/3) but is labeled as "limit". The correct rule name from the allowed vocabulary would be "substitution". This labeling mismatch is a style defect.qwen3.6:27b-mlx: pass — The solution correctly applies algebraic simplification, factoring, and cancellation to resolve the indeterminate form. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies algebraic simplification, factoring, and cancellation to resolve the indeterminate form. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (style) 2026-10-05 — Step 5 applies a substitution (plugging in x=2/3) but is labeled as "limit". The correct rule name from the allowed vocabulary would be "substitution". This labeling mismatch is a style defect.qwen3.6:27b-mlx: fail (style) 2026-10-05 — Step 7 is redundant and mislabeled; converting 1/2 to 0.5 is a simplification or rewrite, not a limit evaluation, and the limit was already evaluated in Step 5.gpt-oss:20b: fail (style) 2026-10-05 — Step 7 applies the label "limit" to a constant value 0.5, which is unnecessary and misleading; the value is already the limit from step 5. Also step 6 is redundant, as step 5 already gives the exact limit.
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 2026-10-05 with SymPy 1.14.0.