Limit of \( \displaystyle 9 x^{2} + 9 x - 1 \) as \( x \to \frac{2}{3} \)
Problem 1.61 · medium
Evaluate \( \displaystyle \lim_{x \to \frac{2}{3}} 9 x^{2} + 9 x - 1 \).
- \[ \lim_{x \to \frac{2}{3}^+}\left(9 x^{2} + 9 x - 1\right) \]limitIdentify the limit to be evaluated.✓ Proved
- \[ = 9 \]limitSince the function is a polynomial, it is continuous, so we can evaluate by direct substitution.✓ Proved
Answer \( 9 \)
✓ 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 |
| 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
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-30gpt-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.