Limit of \( \displaystyle \frac{- 2 x + \left(2 x + 1\right)^{2} - 1}{\left(2 x + 1\right)^{2} - 1} \) as \( x \to 0 \)
Problem 1.41 · hard Video
Evaluate \( \displaystyle \lim_{x \to 0} \frac{- 2 x + \left(2 x + 1\right)^{2} - 1}{\left(2 x + 1\right)^{2} - 1} \).
- \[ \lim_{x \to 0^+}\left(\frac{- 2 x + \left(2 x + 1\right)^{2} - 1}{\left(2 x + 1\right)^{2} - 1}\right) \]limitStart with the limit expression.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{4 x^{2} + 2 x}{4 x^{2} + 4 x}\right) \]algebra simplify simplifyExpand the squared terms. Combine like terms in the numerator and denominator. Combine like terms in the numerator.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{4 x + 2}{4 x + 4}\right) \]factor cancelFactor out x from both numerator and denominator. Cancel the common factor x.✓ Proved
- \[ = \frac{1}{2} \]limitEvaluate the limit by substituting x = 0.✓ Proved
Answer \( \frac{1}{2} \)
✓ 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. How we verify
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 (2*x + 1)**2 - 1 = 0 undefined where 4*x**2 + 4*x = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 + 4*x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 + 4*x = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 + 4*x = 0 undefined where 4*x + 4 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x + 4 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x + 4 = 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, factors, and cancels terms to evaluate the limit. Each step applies a single rule and is labeled correctly.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly expands, simplifies, factors, and cancels terms to evaluate the limit. Each step applies a single rule and is labeled correctly.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies algebraic simplification, factoring, and cancellation to evaluate the limit. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies algebraic simplification, factoring, and cancellation to evaluate the limit. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.gpt-oss:20b: pass 2026-09-29
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.