Home›Calculus 1›Continuity and types of discontinuity›Problem 1.290

Continuity and types of discontinuity

Problem 1.290 · hard

Find every point where \( \displaystyle f(x) = \frac{- 12 x - 24}{x^{2} - 4} \) 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.
Reviewed
\[ x^{2} - 4 = \left(x - 2\right) \left(x + 2\right) \]
Factor the denominator.✓ Proved
\[ \frac{- 12 x - 24}{x^{2} - 4} = - \frac{12}{x - 2} \]
The factor (x + 2) cancels.✓ Proved
\[ \lim_{x \to -2^+}\left(\frac{- 12 x - 24}{x^{2} - 4}\right) = 3 \]
At x = -2 the limit exists, but f(-2) is undefined: a removable discontinuity.✓ Proved
\[ \left. -12 \right|_{\substack{ x=2 }} = -12 \]
At x = 2 the numerator is not 0 while the denominator is, so |f(x)| grows without bound: an infinite discontinuity.✓ Proved

Answer \( \text{removable at } x = -2;\ \text{infinite at } x = 2 \)

✓ 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.

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	claude-sonnet-5-5	claude-sonnet-5-5: This is a true sentence, not an algebraic equivalence: a rational function is continuous wherever its denominator is nonzero, so only zeros of x^2-4 can be discontinuities.
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', 2: 'infinite'}

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — Step 5 incorrectly substitutes x=2 into the simplified numerator -12 to claim the numerator is non-zero, but the original numerator -12(2)-24 equals -48, not 0. While the conclusion of an infinite discontinuity is correct, the reasoning that 'the numerator is not 0' based on the value -12 is factually wrong regarding the original function's numerator at that point, and confusingly mixes the simplified form with the original function's properties.

Senior review claude-sonnet-5-5, 2026-10-05: pass — The answer is correct: f = -12/(x-2) away from x = ±2, the limit at -2 is 3 (removable), and x = 2 gives a nonzero numerator over zero (infinite). Line 5's Subs(-12, x, 2) = -12 is a vacuous way to show the numerator is nonzero, and line 4 shows only a one-sided limit; both are stylistic.

gpt-oss:20b: dismiss — Line 5 substitutes into the constant numerator -12 of the reduced form -12/(x-2), so Subs(-12, x, 2) = -12 is true; the original numerator -48 is also nonzero, so the infinite-discontinuity conclusion holds either way.

Every verdict on record (5)

qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 5 incorrectly substitutes x=2 into the simplified numerator -12 to claim the numerator is non-zero, but the original numerator -12(2)-24 equals -48, not 0. While the conclusion of an infinite discontinuity is correct, the reasoning that 'the numerator is not 0' based on the value -12 is factually wrong regarding the original function's numerator at that point, and confusingly mixes the simplified form with the original function's properties.
gpt-oss:20b: pass 2026-10-05
claude-sonnet-5-5: pass 2026-10-05 — The answer is correct: f = -12/(x-2) away from x = ±2, the limit at -2 is 3 (removable), and x = 2 gives a nonzero numerator over zero (infinite). Line 5's Subs(-12, x, 2) = -12 is a vacuous way to show the numerator is nonzero, and line 4 shows only a one-sided limit; both are stylistic.
qwen3.6:27b-mlx: pass 2026-10-05
gpt-oss:20b: fail (error) 2026-10-05 — The statement ‘Subs(-12, x, 2) = -12’ is incorrect; the numerator at x=2 is -48, not -12. The conclusion about the infinite discontinuity is correct, but the sentence contains a computational error that could mislead a student.

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 2026-10-05 with SymPy 1.14.0.