Limit of \( \displaystyle \frac{4 x^{2} - 1}{2 x - 1} \) as \( x \to \frac{1}{2} \)
Problem 1.162 · medium
Evaluate \( \displaystyle \lim_{x \to \frac{1}{2}} \frac{4 x^{2} - 1}{2 x - 1} \).
- \[ \lim_{x \to \frac{1}{2}^+}\left(\frac{4 x^{2} - 1}{2 x - 1}\right) \]limitStart with the limit of the given function as x approaches 1/2 from the right.✓ Proved
- \[ = \lim_{x \to \frac{1}{2}^+}\left(2 x + 1\right) \]factor cancelFactor the numerator using the difference of squares. Cancel the common factor (2*x - 1) in the numerator and denominator.✓ Proved
- \[ = 2 \]limit simplifyEvaluate the limit by substituting x = 1/2 into the remaining expression. Simplify the final numerical expression.✓ Proved
Answer \( 2 \)
✓ 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 2*x - 1 = 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 |
| 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 factors the numerator, cancels the common term, and evaluates the limit. Each step applies a single rule from the allowed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly factors the numerator, cancels the common term, and evaluates the limit. Each step applies a single rule from the allowed vocabulary.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly factors the numerator, cancels the common term, and evaluates the limit. Each step applies exactly one rule from the allowed 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.