Limit of \( \displaystyle \frac{\ln{\left(x - 1 \right)}}{x - 2} \) as \( x \to 2 \)
Problem 1.428 · easy
Evaluate \( \displaystyle \lim_{x \to 2} \frac{\ln{\left(x - 1 \right)}}{x - 2} \).
- \[ \lim_{x \to 2^+}\left(\frac{\ln{\left(x - 1 \right)}}{x - 2}\right) \]limitIdentify the limit form.✓ Proved
- \[ = \lim_{x \to 2^+}\left(\frac{\frac{d}{d x} \ln{\left(x - 1 \right)}}{\frac{d}{d x} \left(x - 2\right)}\right) \]lhopitalApply L'Hopital's rule because the limit is of the form 0/0.✓ Proved
- \[ = \lim_{x \to 2^+} \frac{1}{x - 1} \]simplify simplifyCompute the derivatives. Simplify the expression.✓ Proved
- \[ = 1 \]limitEvaluate the limit.✓ Proved
Answer \( 1 \)
✓ 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 log is undefined for non-positive arguments undefined where x - 2 = 0 undefined where Derivative(x - 2, x) = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where Derivative(x - 2, x) = 0 undefined where x - 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 1 = 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
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the 0/0 indeterminate form, applies L'Hopital's rule with correct derivatives, and evaluates the resulting limit. Each step changes only one aspect of the expression and uses valid labels.gpt-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.