Limit of \( \displaystyle \frac{2 x + 2}{2 x - 2} \) as \( x \to 0 \)
Problem 1.130 · medium
Evaluate \( \displaystyle \lim_{x \to 0} \frac{2 x + 2}{2 x - 2} \).
- \[ \lim_{x \to 0^+}\left(\frac{2 x + 2}{2 x - 2}\right) \]limit factorStart with the original limit expression. Factor out a 2 from both the numerator and the denominator.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{x + 1}{x - 1}\right) \]simplifyCancel the common factor of 2.✓ Proved
- \[ = -1 \]limitEvaluate the expression by substituting x = 0.✓ 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 undefined where 2*x - 2 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 2 = 0 undefined where x - 1 = 0 |
| 4 | ✓ 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 — The solution correctly factors, simplifies, and evaluates the limit. Each step applies a single rule and uses an appropriate label from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly factors, simplifies, and evaluates the limit. Each step applies a single rule and uses an appropriate label from the fixed vocabulary.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly factors, simplifies, and evaluates the limit. Each step applies a single rule and uses valid labels from the vocabulary.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.