Limit of \( \displaystyle 9 x + \left(3 x - 1\right)^{2} - 4 \) as \( x \to 1 \)
Problem 1.233 · medium
Evaluate \( \displaystyle \lim_{x \to 1} 9 x + \left(3 x - 1\right)^{2} - 4 \).
- \[ \lim_{x \to 1^+}\left(9 x + \left(3 x - 1\right)^{2} - 4\right) \]limitStart with the limit of the function as x approaches 1.✓ Proved
- \[ = \lim_{x \to 1^+}\left(9 x^{2} + 3 x - 3\right) \]simplify simplifyExpand the squared binomial. Combine like terms.✓ Proved
- \[ = 9 \]substitution algebra algebraSubstitute x = 1 into the expression. Evaluate the powers. Perform the final addition and subtraction.✓ 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 |
| 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 |
| 6 | ✓ 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 expands, simplifies, and substitutes to evaluate 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 expands, simplifies, and substitutes to evaluate 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 2026-10-05 — The solution correctly expands, simplifies, and evaluates the limit using valid single-step transformations with appropriate labels.gpt-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.