Limit of \( \displaystyle 6 x + \left(2 x + 1\right)^{2} + 2 \) as \( x \to \frac{1}{2} \)
Problem 1.158 · medium
Evaluate \( \displaystyle \lim_{x \to \frac{1}{2}} 6 x + \left(2 x + 1\right)^{2} + 2 \).
- \[ \lim_{x \to \frac{1}{2}^+}\left(6 x + \left(2 x + 1\right)^{2} + 2\right) \]limitStart with the original limit expression.✓ Proved
- \[ = \lim_{x \to \frac{1}{2}^+} 9 \]substitution algebra algebra algebra algebraSubstitute the value x = 1/2 into the expression. Simplify the multiplication 6 * 1/2. Simplify the expression inside the parentheses. Evaluate the exponent. Sum the remaining terms.✓ Proved
- \[ = 9 \]limitThe limit evaluates to 9.✓ 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 |
| 7 | ✓ 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 applies substitution and arithmetic simplification in single-rule steps. The logic is sound and the labels are appropriate.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies substitution and arithmetic simplification in single-rule steps. The logic is sound and the labels are appropriate.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies substitution and arithmetic simplification steps. Each step changes only one thing and uses valid labels from the fixed vocabulary.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.