Limit of \( \displaystyle \frac{3 x + 1}{3 x - 3} \) as \( x \to \frac{1}{3} \)
Problem 1.228 · medium
Evaluate \( \displaystyle \lim_{x \to \frac{1}{3}} \frac{3 x + 1}{3 x - 3} \).
- \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{3 x + 1}{3 x - 3}\right) \]limitStart with the limit of the function as x approaches 1/3.✓ Proved
- \[ = \lim_{x \to \frac{1}{3}^+}\left(3 x + 1\right) \left(\lim_{x \to \frac{1}{3}^+}\left(3 x - 3\right)\right)^{-1} \]limit-lawApply the quotient rule for limits.✓ Proved
- \[ = \lim_{x \to \frac{1}{3}^+} \tilde{\infty} \]simplifyEvaluate the numerator and denominator at x = 1/3.Not checked
- \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{4}{3 x - 3}\right) \]factorFactor the denominator to analyze the behavior near the singularity.Not checked
- \[ = \infty \]infinityAs x approaches 1/3 from the right, the expression approaches infinity.Not checked
Answer \( -1 \)
Lines: 2 proved, 4 not checked. 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 - 3 = 0 undefined where Limit(3*x - 3, x, 1/3, dir='+') = 0 |
| 3 | Not checked | — | simplify left zoo; no point in the sample was defined on both lines undefined where Limit(3*x - 3, x, 1/3, dir='+') = 0 |
| 4 | Not checked | — | simplify left zoo; no point in the sample was defined on both lines undefined where 3*x - 3 = 0 |
| 5 | Not checked | — | simplify left -oo; no point in the sample was defined on both lines undefined where 3*x - 3 = 0 |
| answer | Not checked | — | final line against the stated answer: simplify left oo; no point in the sample was defined on both lines |
| 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 (error) — Step 3 incorrectly evaluates the numerator and denominator at x=1/3, giving 4/0 instead of the correct 2/(-2). This misapplication of the limit law leads to an incorrect conclusion.qwen3.6:27b-mlx: fail (error) — The stated answer is -1, but the solution concludes with infinity. Additionally, Step 3 incorrectly simplifies the limit expression to a constant 4/0 instead of evaluating the limits of the numerator and denominator separately as required by the quotient rule applied in Step 2.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The stated answer is -1, but the solution concludes with infinity. Additionally, Step 3 incorrectly simplifies the limit expression to a constant 4/0 instead of evaluating the limits of the numerator and denominator separately as required by the quotient rule applied in Step 2.gpt-oss:20b: fail (error) 2026-10-05 — Step 3 incorrectly evaluates the numerator and denominator at x=1/3, giving 4/0 instead of the correct 2/(-2). This misapplication of the limit law leads to an incorrect conclusion.qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 3 incorrectly simplifies the expression to 4/0, which is not a valid algebraic form for further limit analysis; it conflates evaluation with simplification. Step 5 concludes the limit is infinity, but the limit does not exist because the left-hand limit is -infinity and the right-hand limit is +infinity.gpt-oss:20b: fail (error) 2026-10-05 — Step 2 incorrectly applies the quotient rule for limits when the denominator’s limit is zero; the rule requires a non‑zero denominator limit. Step 3 also mis‑evaluates the numerator: 3*(1/3)+1 = 2, not 4. These errors invalidate the solution.
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.