Continuity and types of discontinuity
Problem 1.756 · hard
Find every point where \( \displaystyle f(x) = \frac{x^{2} + x - 2}{x^{2} - 3 x - 10} \) 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.Reviewed
- \[ x^{2} - 3 x - 10 = \left(x - 5\right) \left(x + 2\right) \]Factor the denominator.✓ Proved
- \[ \frac{x^{2} + x - 2}{x^{2} - 3 x - 10} = \frac{x - 1}{x - 5} \]The factor (x + 2) cancels.✓ Proved
- \[ \lim_{x \to -2^+}\left(\frac{x^{2} + x - 2}{x^{2} - 3 x - 10}\right) = \frac{3}{7} \]At x = -2 the limit exists, but f(-2) is undefined: a removable discontinuity.✓ Proved
- \[ \left. x - 1 \right|_{\substack{ x=5 }} = 4 \]At x = 5 the numerator is not 0 while the denominator is, so |f(x)| grows without bound: an infinite discontinuity.✓ Proved
Answer \( \text{removable at } x = -2;\ \text{infinite at } x = 5 \)
✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: {-2: 'removable', 5: 'infinite'} |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the discontinuities by factoring the denominator and analyzing the limits. The classification of x = -2 as removable and x = 5 as infinite is accurate based on the behavior of the simplified function and the original undefined points.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-25 — The solution correctly identifies the discontinuities by factoring the denominator and analyzing the limits. The classification of x = -2 as removable and x = 5 as infinite is accurate based on the behavior of the simplified function and the original undefined points.gpt-oss:20b: pass 2026-10-25qwen3.6:27b-mlx: pass 2026-10-24gpt-oss:20b: pass 2026-10-24qwen3.6:27b-mlx: pass 2026-10-24gpt-oss:20b: pass 2026-10-24
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 2026-10-25 with SymPy 1.14.0.