Continuity and types of discontinuity
Problem 1.541 · hard
Find every point where \( \displaystyle f(x) = \frac{6 - 6 x}{x^{2} + 3 x - 4} \) 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.
- \[ x^{2} + 3 x - 4 = \left(x - 1\right) \left(x + 4\right) \]Factor the denominator.✓ Proved
- \[ \frac{6 - 6 x}{x^{2} + 3 x - 4} = - \frac{6}{x + 4} \]The factor (x − 1) cancels.✓ Proved
- \[ \lim_{x \to 1^+}\left(\frac{6 - 6 x}{x^{2} + 3 x - 4}\right) = - \frac{6}{5} \]At x = 1 the limit exists, but f(1) is undefined: a removable discontinuity.✓ Proved
- \[ \left. -6 \right|_{\substack{ x=-4 }} = -6 \]At x = -4 the numerator is not 0 while the denominator is, so |f(x)| grows without bound: an infinite discontinuity.✓ Proved
Answer \( \text{removable at } x = 1;\ \text{infinite at } x = -4 \)
Lines: 4 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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: {-4: 'infinite', 1: 'removable'} |
Reviewers
gpt-oss:20b: fail (error) — The solution incorrectly evaluates the numerator at x = -4 (it should be 30, not -6) and therefore misstates the reasoning for the infinite discontinuity. The conclusion about the infinite discontinuity is correct, but the justification is wrong.qwen3.6:27b-mlx: fail (error) — Step 5 incorrectly evaluates the numerator of the simplified function at x = -4 as -6, whereas -6/(-4 + 4) involves division by zero. The numerator of the original function at x = -4 is 30, not -6. While the conclusion of an infinite discontinuity is correct, the reasoning provided in the sentence is mathematically false and misleading.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-11 — Step 5 incorrectly evaluates the numerator of the simplified function at x = -4 as -6, whereas -6/(-4 + 4) involves division by zero. The numerator of the original function at x = -4 is 30, not -6. While the conclusion of an infinite discontinuity is correct, the reasoning provided in the sentence is mathematically false and misleading.gpt-oss:20b: fail (error) 2026-10-11 — The solution incorrectly evaluates the numerator at x = -4 (it should be 30, not -6) and therefore misstates the reasoning for the infinite discontinuity. The conclusion about the infinite discontinuity is correct, but the justification is wrong.qwen3.6:27b-mlx: fail (error) 2026-10-11 — Step 5 incorrectly substitutes x = -4 into the simplified numerator -6, which is non-zero, but fails to explicitly state that the denominator is zero at x = -4. While the conclusion is correct, the reasoning 'Subs(-6, x, -4) = -6' is algebraically trivial and does not constitute a valid argument for an infinite discontinuity without referencing the denominator's behavior. A student might incorrectly conclude that any non-zero numerator implies an infinite discontinuity regardless of the denominator.gpt-oss:20b: fail (error) 2026-10-11 — Step 5 incorrectly evaluates the numerator at x = –4; it should be 6 – 6(–4) = 30, not –6. The conclusion about an infinite discontinuity is correct, but the calculation is wrong.
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 2026-10-11 with SymPy 1.14.0.