Limit of \( \displaystyle \frac{9 x^{2} - 4}{3 x - 2} \) as \( x \to \frac{2}{3} \)
Problem 1.241 · medium
Evaluate \( \displaystyle \lim_{x \to \frac{2}{3}} \frac{9 x^{2} - 4}{3 x - 2} \).
- \[ \lim_{x \to \frac{2}{3}^+}\left(\frac{9 x^{2} - 4}{3 x - 2}\right) \]limitSet up the one-sided limit as specified.✓ Proved
- \[ = \lim_{x \to \frac{2}{3}^+}\left(3 x + 2\right) \]factor cancelFactor the numerator using the difference of squares. Cancel the common factor (3*x - 2) in the numerator and denominator.✓ Proved
- \[ = 4 \]limit simplifyEvaluate the limit by substituting x = 2/3. Perform the final arithmetic.✓ 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.
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 |
| 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 factors the numerator, cancels the common term, and evaluates the limit. Each step applies a single rule and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly factors the numerator, cancels the common term, and evaluates the limit. Each step applies a single rule and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05gpt-oss:20b: pass 2026-10-05
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.