Continuity and types of discontinuity
Problem 1.759 · hard
Find every point where \( \displaystyle f(x) = \frac{9 - 3 x}{x^{2} + x - 12} \) 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.
- \[ x^{2} + x - 12 = \left(x - 3\right) \left(x + 4\right) \]Factor the denominator.✓ Proved
- \[ \frac{9 - 3 x}{x^{2} + x - 12} = - \frac{3}{x + 4} \]The factor (x − 3) cancels.✓ Proved
- \[ \lim_{x \to 3^+}\left(\frac{9 - 3 x}{x^{2} + x - 12}\right) = - \frac{3}{7} \]At x = 3 the limit exists, but f(3) is undefined: a removable discontinuity.✓ Proved
- \[ \left. -3 \right|_{\substack{ x=-4 }} = -3 \]At x = -4 the numerator is not 0 while the denominator is, so |f(x)| grows without bound: an infinite discontinuity.✓ Proved
Answer \( \text{removable at } x = 3;\ \text{infinite at } x = -4 \)
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. How we verify
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: {-4: 'infinite', 3: 'removable'} |
Reviewers
gpt-oss:20b: fail (misleading) — The solution incorrectly evaluates the function at x = -4 by substituting into the simplified form, which still has a denominator of zero. It should state that the numerator is non‑zero while the denominator vanishes, leading to an infinite discontinuity, rather than giving an incorrect finite value.qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution fails to classify the discontinuity at x = -4 correctly. While it is indeed an infinite discontinuity, the limit is not just 'growing without bound' in magnitude; the one-sided limits are +infinity and -infinity, which is the definition of an infinite discontinuity. More critically, step 5 uses 'Subs(-3, x, -4)' which is nonsensical notation for evaluating a limit or checking the numerator of the simplified function. The numerator of the simplified function -3/(x+4) is -3, which is non-zero, confirming the infinite discontinuity, but the notation 'Subs(-3, x, -4)' is mathematically ill-defined and misleading. A better explanation would evaluate the limit of the simplified function as x approaches -4.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-25 — [domain objection, downgraded to style] The solution fails to classify the discontinuity at x = -4 correctly. While it is indeed an infinite discontinuity, the limit is not just 'growing without bound' in magnitude; the one-sided limits are +infinity and -infinity, which is the definition of an infinite discontinuity. More critically, step 5 uses 'Subs(-3, x, -4)' which is nonsensical notation for evaluating a limit or checking the numerator of the simplified function. The numerator of the simplified function -3/(x+4) is -3, which is non-zero, confirming the infinite discontinuity, but the notation 'Subs(-3, x, -4)' is mathematically ill-defined and misleading. A better explanation would evaluate the limit of the simplified function as x approaches -4.gpt-oss:20b: fail (misleading) 2026-10-25 — The solution incorrectly evaluates the function at x = -4 by substituting into the simplified form, which still has a denominator of zero. It should state that the numerator is non‑zero while the denominator vanishes, leading to an infinite discontinuity, rather than giving an incorrect finite value.qwen3.6:27b-mlx: fail (error) 2026-10-24 — Step 5 incorrectly substitutes x = -4 into the simplified expression -3/(x+4), resulting in a division by zero error rather than evaluating the numerator. The correct reasoning is that the limit of the simplified function as x approaches -4 is infinite because the denominator approaches 0 while the numerator approaches -7.gpt-oss:20b: fail (style) 2026-10-24 — [domain objection, downgraded to style] Sentence 5 incorrectly states that substituting x = -4 into the simplified expression yields -3; the correct evaluation is undefined, but the conclusion that the discontinuity is infinite is still correct.qwen3.6:27b-mlx: fail (style) 2026-10-24 — [domain objection, downgraded to style] Step 5 incorrectly substitutes x = -4 into the simplified expression -3/(x+4), which is undefined at x = -4. The classification of the discontinuity as infinite is correct, but the reasoning provided is mathematically invalid and misleading.gpt-oss:20b: fail (style) 2026-10-24 — [domain objection, downgraded to style] The solution incorrectly states that Subs(-3, x, -4) = -3, which is false; the correct evaluation of the simplified expression at x = -4 is undefined, leading to an infinite discontinuity. The conclusion about the infinite discontinuity is correct, but the sentence misleads the reader.
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.