Limit of \( \displaystyle \frac{4 x^{2} - 2 x}{4 x^{2} - 1} \) as \( x \to \frac{1}{2} \)
Problem 1.414 · medium
Evaluate \( \displaystyle \lim_{x \to \frac{1}{2}} \frac{4 x^{2} - 2 x}{4 x^{2} - 1} \).
- \[ \lim_{x \to \frac{1}{2}^+}\left(\frac{4 x^{2} - 2 x}{4 x^{2} - 1}\right) \]limitStart with the limit of the function as x approaches 1/2 from the right.✓ Proved
- \[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{2 x \left(2 x - 1\right)}{4 x^{2} - 1}\right) \]factorFactor the numerator.✓ Proved
- \[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{2 x}{2 x + 1}\right) \]factor cancelFactor the denominator using the difference of squares. Cancel the common factor (2*x - 1).✓ Proved
- \[ = \frac{1}{2} \]limit simplifyEvaluate the limit by substituting x = 1/2. Simplify the resulting fraction.✓ 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.
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 4*x**2 - 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 1 = 0 undefined where 2*x + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 1 = 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: fail (style) — Step 5 applies a substitution (x=1/2) to evaluate the limit, but it is labeled as "limit". The correct label for this step should be "substitution".qwen3.6:27b-mlx: pass — The solution correctly factors the numerator and denominator, cancels the common term, and evaluates the limit by substitution. Each step applies exactly one rule from the allowed vocabulary.
Every verdict on record (4)
gpt-oss:20b: fail (style) 2026-10-08 — Step 5 applies a substitution (x=1/2) to evaluate the limit, but it is labeled as "limit". The correct label for this step should be "substitution".qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly factors the numerator and denominator, cancels the common term, and evaluates the limit by substitution. Each step applies exactly one rule from the allowed vocabulary.qwen3.6:27b-mlx: pass 2026-10-08gpt-oss:20b: pass 2026-10-08
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 2026-10-08 with SymPy 1.14.0.