Continuity and types of discontinuity
Problem 1.195 · hard
Find every point where \( \displaystyle f(x) = \frac{2 x^{2} - 2}{x^{2} - x - 2} \) is discontinuous, and classify each discontinuity as removable, a jump, or infinite.
- A rational function is continuous wherever its denominator is not zero, so only the zeros of the denominator can be discontinuities.
- \[ x^{2} - x - 2 = \left(x - 2\right) \left(x + 1\right) \]Factor the denominator.✓ Proved
- \[ \frac{2 x^{2} - 2}{x^{2} - x - 2} = \frac{2 x - 2}{x - 2} \]The factor (x + 1) cancels.✓ Proved
- \[ \lim_{x \to -1^+}\left(\frac{2 x^{2} - 2}{x^{2} - x - 2}\right) = \frac{4}{3} \]At x = -1 the limit exists, but f(-1) is undefined: a removable discontinuity.✓ Proved
- \[ \left. 2 x - 2 \right|_{\substack{ x=2 }} = 2 \]At x = 2 the numerator is not 0 while the denominator is, so |f(x)| grows without bound: an infinite discontinuity.✓ Proved
Answer \( \text{removable at } x = -1;\ \text{infinite at } x = 2 \)
Lines: 4 proved, 1 not checked. 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 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | zeros of the denominator found by solve, each probed at ±1e-9: {-1: 'removable', 2: 'infinite'} |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly claims that the limit at x = -1 is 4/3; the correct limit is 4/3 only if the simplified function is (2x-2)/(x-2) evaluated a
Every verdict on record (4)
qwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly claims that the limit at x = -1 is 4/3; the correct limit is 4/3 only if the simplified function is (2x-2)/(x-2) evaluated agpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the points of discontinuity, simplifies the function to classify them, and applies the correct definitions for removable and infinite discontinuities.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
generator:structured/continuity_classify, checked 2026-10-04 with SymPy 1.14.0.