Limit of \( \displaystyle \frac{\ln{\left(3 x - 1 \right)}}{3 x - 2} \) as \( x \to \frac{2}{3} \)
Problem 1.240 · medium
Evaluate \( \displaystyle \lim_{x \to \frac{2}{3}} \frac{\ln{\left(3 x - 1 \right)}}{3 x - 2} \).
- \[ \lim_{x \to \frac{2}{3}^+}\left(\frac{\ln{\left(3 x - 1 \right)}}{3 x - 2}\right) \]limitStart with the limit of the given function.✓ Proved
- \[ = \lim_{x \to \frac{2}{3}^+}\left(\frac{\tilde{\infty}}{3 x - 2}\right) \]simplifyEvaluate the numerator at x = 2/3.Not checked
- \[ = \lim_{x \to \frac{2}{3}^+}\left(- \frac{\infty}{3 x - 2}\right) \]simplifyNote that log(0) approaches negative infinity.Not checked
- \[ = \lim_{x \to \frac{2}{3}^+} \infty \]simplify simplifyEvaluate the denominator at x = 2/3. Simplify the fraction involving infinity.Not checked
- \[ = \infty \]limitThe limit evaluates to infinity.Not checked
Answer \( 1 \)
Lines: 1 proved, 6 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | Not checked | — | simplify left 1 - Limit(zoo/(3*x - 2), x, 2/3, dir='+'); no point in the sample was defined on both lines log is undefined for non-positive arguments undefined where 3*x - 2 = 0 |
| 3 | Not checked | — | simplify left Limit(zoo/(3*x - 2), x, 2/3, dir='+') - Limit(-oo/(3*x - 2), x, 2/3, dir='+'); no point in the sample was defined on both lines undefined where 3*x - 2 = 0 |
| 4 | Not checked | — | simplify left -oo + Limit(-oo/(3*x - 2), x, 2/3, dir='+'); no point in the sample was defined on both lines undefined where 3*x - 2 = 0 |
| 5 | Not checked | — | simplify left nan; no point in the sample was defined on both lines |
| 6 | Not checked | — | simplify left nan; no point in the sample was defined on both lines |
| 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 4 incorrectly evaluates the denominator 3*x-2 at x=2/3; it equals 0, not -4/3. Consequently, the subsequent simplifications involving division by zero and infinity are invalid.qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution incorrectly evaluates the limit as infinity, whereas the correct limit is 1 (since the argument of the log approaches 0, making the numerator approach 0, and the denominator approaches -4/3, resulting in 0 / (-4/3) = 0, wait, log(0) is -infinity? No, log(3*(2/3)-1) = log(0) which is undefined/infinity, but the limit of log(u) as u->0+ is -infinity. So the numerator goes to -infinity and the denominator goes to -4/3. The ratio is infinity. Wait, let me re-read the problem. log(3x-1)/(3x-2). As x->2/3, 3x-1 -> 0. log(0) -> -infinity. 3x-2 -> -4/3. So the limit is -infinity / (-4/3) = +infinity. The stated answer is 1. The steps conclude oo. The stated answer is 1. The steps are internally consistent but the final answer 1 is wrong? Or is the problem log(3x-1) / (3x-2) or log( (3x-1)/(3x-2) )? The text says log(3*x - 1)/(3*x - 2). Usually this means (log(3x-1))/(3x-2). If it meant log((3x-1)/(3x-2)), the limit would be log(0/-4/3) = log(0) = -infinity. If it meant log(3x-1) / (3x-2), the limit is infinity. The stated answer is 1. This implies the problem might have been different, e.g., log(3x-1)/(x-1/3) or similar. However, I must judge the solution against the contract. The contract says 'Stated answer: 1'. The steps conclude 'oo'. The steps do not match the stated answer. Also, step 2 'Evaluate the numerator at x=2/3' is not a valid limit operation; you cannot substitute into a limit if it results in an indeterminate form or singularity without justification, but here it's a direct substitution into a divergent part. More importantly, the steps conclude with 'oo', but the stated answer is '1'. This is a contradiction between the steps and the stated answer. Additionally, the logic in step 2-5 is flawed because you cannot simply 'evaluate' parts of a limit expression to infinity and then simplify algebraically with infinity in this manner without proper limit laws (like limit of product/quotient). Specifically, step 2 replaces log(3x-1) with log(0) inside the limit, which is not standard notation. Step 3 replaces log(0) with -oo. This is a heuristic, not a rigorous step. But the biggest issue is that the steps conclude 'oo' while the stated answer is '1'. This is a failure to match the stated answer. Also, is the limit actually 1? Let's check. If the problem was log(3x-1)/(3x-2), limit is infinity. If the problem was log(3x-1)/(x-1/3), limit is 3. If the problem was log(3x-2)/(x-2/3), limit is 3. If the problem was log(3x-1)/(x-1/3) ... wait. Maybe the problem is log(3x-1) / (x - 1/3)? No, the text is explicit. Let's assume the text is correct. The limit is infinity. The stated answer 1 is wrong. The steps conclude infinity. So the steps are 'correct' for the expression, but the stated answer is wrong. The prompt asks to review the solution. The solution includes the steps and the stated answer. They contradict. This is an error.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-05 — [domain objection, downgraded to style] The solution incorrectly evaluates the limit as infinity, whereas the correct limit is 1 (since the argument of the log approaches 0, making the numerator approach 0, and the denominator approaches -4/3, resulting in 0 / (-4/3) = 0, wait, log(0) is -infinity? No, log(3*(2/3)-1) = log(0) which is undefined/infinity, but the limit of log(u) as u->0+ is -infinity. So the numerator goes to -infinity and the denominator goes to -4/3. The ratio is infinity. Wait, let me re-read the problem. log(3x-1)/(3x-2). As x->2/3, 3x-1 -> 0. log(0) -> -infinity. 3x-2 -> -4/3. So the limit is -infinity / (-4/3) = +infinity. The stated answer is 1. The steps conclude oo. The stated answer is 1. The steps are internally consistent but the final answer 1 is wrong? Or is the problem log(3x-1) / (3x-2) or log( (3x-1)/(3x-2) )? The text says log(3*x - 1)/(3*x - 2). Usually this means (log(3x-1))/(3x-2). If it meant log((3x-1)/(3x-2)), the limit would be log(0/-4/3) = log(0) = -infinity. If it meant log(3x-1) / (3x-2), the limit is infinity. The stated answer is 1. This implies the problem might have been different, e.g., log(3x-1)/(x-1/3) or similar. However, I must judge the solution against the contract. The contract says 'Stated answer: 1'. The steps conclude 'oo'. The steps do not match the stated answer. Also, step 2 'Evaluate the numerator at x=2/3' is not a valid limit operation; you cannot substitute into a limit if it results in an indeterminate form or singularity without justification, but here it's a direct substitution into a divergent part. More importantly, the steps conclude with 'oo', but the stated answer is '1'. This is a contradiction between the steps and the stated answer. Additionally, the logic in step 2-5 is flawed because you cannot simply 'evaluate' parts of a limit expression to infinity and then simplify algebraically with infinity in this manner without proper limit laws (like limit of product/quotient). Specifically, step 2 replaces log(3x-1) with log(0) inside the limit, which is not standard notation. Step 3 replaces log(0) with -oo. This is a heuristic, not a rigorous step. But the biggest issue is that the steps conclude 'oo' while the stated answer is '1'. This is a failure to match the stated answer. Also, is the limit actually 1? Let's check. If the problem was log(3x-1)/(3x-2), limit is infinity. If the problem was log(3x-1)/(x-1/3), limit is 3. If the problem was log(3x-2)/(x-2/3), limit is 3. If the problem was log(3x-1)/(x-1/3) ... wait. Maybe the problem is log(3x-1) / (x - 1/3)? No, the text is explicit. Let's assume the text is correct. The limit is infinity. The stated answer 1 is wrong. The steps conclude infinity. So the steps are 'correct' for the expression, but the stated answer is wrong. The prompt asks to review the solution. The solution includes the steps and the stated answer. They contradict. This is an error.gpt-oss:20b: fail (error) 2026-10-05 — Step 4 incorrectly evaluates the denominator 3*x-2 at x=2/3; it equals 0, not -4/3. Consequently, the subsequent simplifications involving division by zero and infinity are invalid.qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution incorrectly evaluates the limit as infinity by treating the numerator and denominator independently. As x approaches 2/3, the denominator approaches 0, creating an indeterminate form of type infinity/0 (or more precisely, the ratio of a term going to -infinity and a term going to 0). The sign of the denominator changes depending on the direction of approach (from left it is negative, from right it is positive), so the limit does not exist (it is not simply infinity). Specifically, log(3x-1) goes to -infinity. If x -> 2/3+, 3x-2 -> 0+, so ratio -> -infinity. If x -> 2/3-, 3x-2 -> 0-, so ratio -> +infinity. The limit does not exist. The step 'Limit(-oo/(-4/3))' is mathematically invalid because you cannot substitute the limit of the denominator (which is 0) into the expression while the numerator is infinite; this is not a valid application of limit laws for indeterminate forms.gpt-oss:20b: fail (error) 2026-10-05 — The numerator at x=2/3 is log(3*(2/3)-1)=log(1)=0, not log(0). The subsequent steps incorrectly treat log(0) as negative infinity, leading to an incorrect limit evaluation.
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.