Limit of \( \displaystyle \frac{\left(3 x + 1\right)^{2} - 4}{3 x - 1} \) as \( x \to \frac{1}{3} \)
Problem 1.771 · medium Video
Evaluate \( \displaystyle \lim_{x \to \frac{1}{3}} \frac{\left(3 x + 1\right)^{2} - 4}{3 x - 1} \).
- \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{\left(3 x + 1\right)^{2} - 4}{3 x - 1}\right) \]limit simplifyStart with the limit of the function as x approaches 1/3 from the right. Expand the numerator.✓ Proved
- \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{9 x^{2} + 6 x - 3}{3 x - 1}\right) \]simplify factor factorCombine constant terms in the numerator. Factor out the common factor 3 from the numerator. Factor the quadratic expression in the numerator.✓ Proved
- \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{\left(x + 1\right) \left(9 x - 3\right)}{3 x - 1}\right) \]algebraCancel the common factor (3*x - 1) from the numerator and denominator.✓ Proved
- \[ = \lim_{x \to \frac{1}{3}^+}\left(3 x + 3\right) \]limitEvaluate the limit by substituting x = 1/3.✓ Proved
- \[ = 4 \]simplify simplifyPerform the addition inside the parentheses. Final result.✓ Proved
Answer \( 4 \)
✓ 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. How we verify
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 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x - 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x - 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x - 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x - 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x - 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 9 | ✓ 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 6 applies a cancellation of the common factor (3*x-1) but labels the rule as "algebra" instead of the appropriate "cancel". This mislabeling violates the rule‑granularity requirement.qwen3.6:27b-mlx: fail (style) — Step 6 is labeled 'algebra' but performs a cancellation, which should be labeled 'cancel'. Step 5 is labeled 'factor' but the note describes factoring the quadratic, while the step actually shows the result of that factoring; however, the label 'factor' is acceptable for the transformation from step 4 to 5. The primary defect is the mislabeling of the cancellation in step 6.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-26 — Step 6 is labeled 'algebra' but performs a cancellation, which should be labeled 'cancel'. Step 5 is labeled 'factor' but the note describes factoring the quadratic, while the step actually shows the result of that factoring; however, the label 'factor' is acceptable for the transformation from step 4 to 5. The primary defect is the mislabeling of the cancellation in step 6.gpt-oss:20b: fail (style) 2026-10-26 — Step 6 applies a cancellation of the common factor (3*x-1) but labels the rule as "algebra" instead of the appropriate "cancel". This mislabeling violates the rule‑granularity requirement.qwen3.6:27b-mlx: fail (style) 2026-10-25 — Step 6 is labeled 'algebra' but the note says 'Cancel the common factor', which corresponds to the 'cancel' rule in the vocabulary. Additionally, Step 7 is labeled 'limit' but performs substitution, which should be labeled 'substitution'.gpt-oss:20b: fail (style) 2026-10-25 — Step 6 applies a cancellation of the common factor (3*x-1) but labels the operation as "algebra". The correct rule name from the allowed vocabulary is "cancel" (or "simplify" if only simplification is intended).qwen3.6:27b-mlx: fail (style) 2026-10-25 — Step 6 is labeled 'algebra' but the note describes 'Cancel the common factor', which corresponds to the 'cancel' rule in the vocabulary. Additionally, Step 7 is labeled 'limit' but performs substitution, which should be labeled 'substitution'.gpt-oss:20b: pass 2026-10-25
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 2026-10-26 with SymPy 1.14.0.