Limit of \( \displaystyle \frac{\sqrt{2 x - 1} - 3}{2 x - 10} \) as \( x \to 5 \)
Problem 1.557 · medium
Evaluate \( \displaystyle \lim_{x \to 5} \frac{\sqrt{2 x - 1} - 3}{2 x - 10} \).
- \[ \lim_{x \to 5^+}\left(\frac{\sqrt{2 x - 1} - 3}{2 x - 10}\right) \]limit algebraStart with the limit from the right. Multiply numerator and denominator by the conjugate.✓ Proved
- \[ = \lim_{x \to 5^+} \frac{1}{\sqrt{2 x - 1} + 3} \]simplify simplify cancelSimplify the numerator. Combine terms in the numerator. Cancel the common factor (2*x - 10).✓ Proved
- \[ = \lim_{x \to 5^+} \frac{1}{6} \]limitEvaluate the expression at x = 5.✓ Proved
- \[ = \frac{1}{6} \]simplify simplifySimplify the denominator. Final result.✓ 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 - 10 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 10 = 0 undefined where sqrt(2*x - 1) + 3 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sqrt(2*x - 1) + 3 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sqrt(2*x - 1) + 3 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sqrt(2*x - 1) + 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 |
| 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 6 incorrectly applies a limit after substituting the value of x; it should simply evaluate the expression, not keep a Limit object. This mixes substitution and limit evaluation in one step, violating the one‑rule‑per‑step rule.qwen3.6:27b-mlx: pass
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-12 — Step 6 incorrectly applies a limit after substituting the value of x; it should simply evaluate the expression, not keep a Limit object. This mixes substitution and limit evaluation in one step, violating the one‑rule‑per‑step rule.qwen3.6:27b-mlx: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12gpt-oss:20b: pass 2026-10-12
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 2026-10-12 with SymPy 1.14.0.