Limit of \( \displaystyle \frac{\sqrt{\left(2 x - 1\right)^{2} + 9} - 3}{\left(2 x - 1\right)^{2}} \) as \( x \to \frac{1}{2} \)
Problem 1.116 · hard
Evaluate \( \displaystyle \lim_{x \to \frac{1}{2}} \frac{\sqrt{\left(2 x - 1\right)^{2} + 9} - 3}{\left(2 x - 1\right)^{2}} \).
- \[ \lim_{x \to \frac{1}{2}^+}\left(\frac{\sqrt{\left(2 x - 1\right)^{2} + 9} - 3}{\left(2 x - 1\right)^{2}}\right) \]limit rationalizeStart with the original limit expression. Multiply the numerator and denominator by the conjugate of the numerator.✓ Proved
- \[ = \lim_{x \to \frac{1}{2}^+} \frac{1}{\sqrt{\left(2 x - 1\right)^{2} + 9} + 3} \]algebra simplify cancelExpand the numerator. Simplify the expression by subtracting 9. Cancel the common term (2*x - 1)**2.✓ Proved
- \[ = \lim_{x \to \frac{1}{2}^+} \frac{1}{6} \]substitution simplify simplifySubstitute x = 1/2 into the expression. Evaluate the square root. Simplify the denominator.✓ Proved
- \[ = \frac{1}{6} \]limitThe final value of the limit.✓ Proved
Answer \( \frac{1}{6} \)
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. The reviewers disagree about how one step is explained; every verdict is in the receipt.
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 undefined where 2*x - 1 = 0 undefined where sqrt((2*x - 1)**2 + 9) + 3 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sqrt((2*x - 1)**2 + 9) + 3 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sqrt((2*x - 1)**2 + 9) + 3 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sqrt((2*x - 1)**2 + 9) + 3 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 9 | ✓ 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 (error) — Step 7 incorrectly simplifies √((2*0)**2+9)+3 to √(3+3). The correct evaluation is √(0+9)+3 = 3+3 = 6, not √(6). This misstep invalidates the subsequent simplification.qwen3.6:27b-mlx: pass — The solution correctly applies the conjugate method to resolve the indeterminate form. Each step adheres to the single-rule constraint and uses valid labels from the provided vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the conjugate method to resolve the indeterminate form. Each step adheres to the single-rule constraint and uses valid labels from the provided vocabulary.gpt-oss:20b: fail (error) 2026-10-03 — Step 7 incorrectly simplifies √((2*0)**2+9)+3 to √(3+3). The correct evaluation is √(0+9)+3 = 3+3 = 6, not √(6). This misstep invalidates the subsequent simplification.qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies rationalization to resolve the indeterminate form, followed by algebraic simplification and cancellation. The final substitution and evaluation are correct.gpt-oss:20b: pass 2026-10-03
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.