Limit of \( \displaystyle \frac{3 x + 2}{3 x - 2} \) as \( x \to 0 \)
Problem 1.76 · medium
- \[ \lim_{x \to 0^+}\left(\frac{3 x + 2}{3 x - 2}\right) \]limitStart with the original limit expression.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{3 x}{3 x - 2} + \frac{2}{3 x - 2}\right) \]algebraSplit the fraction into two parts.✓ Proved
- \[ = \lim_{x \to 0^+}\left(- \frac{3 x}{2} - 1\right) \]substitutionSubstitute the value x = 0 into the expression.✓ Proved
- \[ = \lim_{x \to 0^+} -1 \]simplifySimplify the terms inside the limit.✓ Proved
- \[ = -1 \]limitEvaluate the final numerical value.✓ Proved
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. A reviewer's objection to the explanation stands; see the receipt.
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 - 2 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x - 2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ 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 3 applies the substitution rule to only the denominator while leaving the numerator as a function of x, violating the single-rule-per-step constraint and resulting in a mathematically invalid expression. Substitution must be applied to the entire expression or handled via limit laws after splitting.
Senior review claude-sonnet-5-5, 2026-10-03: fail — Step 4 is labelled 'simplify', but it replaces the numerator 3*x with 0, which is a substitution. It also evaluates 3*0-2 in the same step, so it does two things under a wrong label. The chain also treats Limit(3x/(-2)-1) as if it were the constant -1 without justification, which teaches sloppy limit handling, even though the final value -1 is correct.
qwen3.6:27b-mlx: dismiss — Substituting x=0 into the denominators only is a legitimate single change, since the contract allows a rule to be applied to part of an expression, and the denominators' limit is nonzero; the real defect is in step 4, not step 3.
Every verdict on record (5)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 3 applies the substitution rule to only the denominator while leaving the numerator as a function of x, violating the single-rule-per-step constraint and resulting in a mathematically invalid expression. Substitution must be applied to the entire expression or handled via limit laws after splitting.gpt-oss:20b: pass 2026-10-03claude-sonnet-5-5: fail (misleading) 2026-10-03 — Step 4 is labelled 'simplify', but it replaces the numerator 3*x with 0, which is a substitution. It also evaluates 3*0-2 in the same step, so it does two things under a wrong label. The chain also treats Limit(3x/(-2)-1) as if it were the constant -1 without justification, which teaches sloppy limit handling, even though the final value -1 is correct.qwen3.6:27b-mlx: fail (error) 2026-09-30 — Step 3 applies substitution to only the denominators while leaving the numerators as variables, which is not a valid application of the substitution rule. Substitution must be applied to the entire expression or consistent sub-expressions, not selectively to parts of a fraction.gpt-oss:20b: pass 2026-09-30
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 2026-10-03 with SymPy 1.14.0.