Home›Calculus 1›Continuity and types of discontinuity›Problem 1.544

Continuity and types of discontinuity

Problem 1.544 · hard

Find every point where \( \displaystyle f(x) = \frac{25 - x^{2}}{x^{2} - x - 20} \) 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
\[ x^{2} - x - 20 = \left(x - 5\right) \left(x + 4\right) \]
Factor the denominator.✓ Proved
\[ \frac{25 - x^{2}}{x^{2} - x - 20} = \frac{- x - 5}{x + 4} \]
The factor (x − 5) cancels.✓ Proved
\[ \lim_{x \to 5^+}\left(\frac{25 - x^{2}}{x^{2} - x - 20}\right) = - \frac{10}{9} \]
At x = 5 the limit exists, but f(5) is undefined: a removable discontinuity.✓ Proved
\[ \left. - x - 5 \right|_{\substack{ x=-4 }} = -1 \]
At x = -4 the numerator is not 0 while the denominator is, so |f(x)| grows without bound: an infinite discontinuity.✓ Proved

Answer \( \text{removable at } x = 5;\ \text{infinite at } x = -4 \)

✓ 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: Line 1 is a sentence, not an equation. It states the standard fact that a rational function is continuous wherever its denominator is nonzero, so only zeros of the denominator can be discontinuities. This is the problem setup and is true.
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', 5: 'removable'}

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-11: pass — The factorization, the cancellation, the removable discontinuity at x=5 (limit -10/9, f(5) undefined) and the infinite discontinuity at x=-4 (original numerator 9, denominator 0) are all correct and match the stated answer. Line 4 uses a one-sided limit, but the two-sided limit is the same, so this is harmless.

qwen3.6:27b-mlx: dismiss — The objection is unparseable and talks itself into the correct value: the limit at x=5 is -(5+5)/(5+4) = -10/9, which matches line 4 and SymPy's evidence.

Every verdict on record (5)

qwen3.6:27b-mlx: pass 2026-10-11
gpt-oss:20b: pass 2026-10-11
claude-sonnet-5-5: pass 2026-10-11 — The factorization, the cancellation, the removable discontinuity at x=5 (limit -10/9, f(5) undefined) and the infinite discontinuity at x=-4 (original numerator 9, denominator 0) are all correct and match the stated answer. Line 4 uses a one-sided limit, but the two-sided limit is the same, so this is harmless.
qwen3.6:27b-mlx: inconclusive 2026-10-11 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The limit calculation at x=5 is incorrect; the limit is -10/9, but the simplified function is (-x-5)/(x+4), so at x=5 it is -10/9. Wait, let me re-ca
gpt-oss:20b: pass 2026-10-11

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.