Limit of \( \displaystyle \frac{3 x}{3 x - 4} \) as \( x \to \frac{2}{3} \)
Problem 1.161 · easy
Evaluate \( \displaystyle \lim_{x \to \frac{2}{3}} \frac{3 x}{3 x - 4} \).
- \[ \lim_{x \to \frac{2}{3}^+}\left(\frac{3 x}{3 x - 4}\right) \]limitStart with the given limit.✓ Proved
- \[ = \lim_{x \to \frac{2}{3}^+} -1 \]substitution simplify simplifySubstitute the value x = 2/3 into the expression. Simplify the numerator and denominator. Perform the division in the denominator.✓ Proved
- \[ = -1 \]limitEvaluate the final numerical value.✓ Proved
Answer \( -1 \)
✓ 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 - 4 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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: pass — The solution correctly identifies that direct substitution is valid because the limit point x=2/3 is not a singularity (the singularity is at x=4/3). The steps are algebraically sound and correctly labeled.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies that direct substitution is valid because the limit point x=2/3 is not a singularity (the singularity is at x=4/3). The steps are algebraically sound and correctly labeled.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies that the limit can be evaluated by direct substitution since the function is continuous at x = 2/3. Each step applies a single rule and is labeled correctly.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.