Home›Calculus 1›Continuity and types of discontinuity›Problem 1.485

Continuity and types of discontinuity

Problem 1.485 · hard

Find every point where \( \displaystyle f(x) = \frac{- 10 x - 30}{x^{2} + 2 x - 3} \) 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
\[ x^{2} + 2 x - 3 = \left(x - 1\right) \left(x + 3\right) \]
Factor the denominator.✓ Proved
\[ \frac{- 10 x - 30}{x^{2} + 2 x - 3} = - \frac{10}{x - 1} \]
The factor (x + 3) cancels.✓ Proved
\[ \lim_{x \to -3^+}\left(\frac{- 10 x - 30}{x^{2} + 2 x - 3}\right) = \frac{5}{2} \]
At x = -3 the limit exists, but f(-3) is undefined: a removable discontinuity.✓ Proved
\[ \left. -10 \right|_{\substack{ x=1 }} = -10 \]
At x = 1 the numerator is not 0 while the denominator is, so |f(x)| grows without bound: an infinite discontinuity.✓ Proved

Answer \( \text{removable at } x = -3;\ \text{infinite at } x = 1 \)

✓ 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 sentence, not an equation. A rational function is continuous wherever its denominator is nonzero, so only the zeros of x^2+2x-3 can be discontinuities. That is a standard true fact.
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: {-3: 'removable', 1: 'infinite'}

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — Line 5 incorrectly substitutes x=1 into the simplified numerator -10 instead of the original numerator -10x-30. While the conclusion (infinite discontinuity) is correct, the justification is mathematically false because the original numerator is -40 at x=1, not -10. This teaches incorrect evaluation techniques.

Senior review claude-sonnet-5-5, 2026-10-09: pass — The factorization, the cancellation, the limit 5/2 at x=-3, and the infinite discontinuity at x=1 are all correct and match the stated answer. Line 5 is a slightly terse way to show the numerator is nonzero, but it is not wrong.

gpt-oss:20b: dismiss — Line 5 evaluates the numerator of the reduced form -10/(x-1), which is the constant -10, so Subs(-10, x, 1) = -10 is true; the original numerator at x=1 is -40, also nonzero, so the conclusion is unaffected and nothing false is stated.

Every verdict on record (5)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: fail (error) 2026-10-09 — Line 5 incorrectly substitutes x=1 into the simplified numerator -10 instead of the original numerator -10x-30. While the conclusion (infinite discontinuity) is correct, the justification is mathematically false because the original numerator is -40 at x=1, not -10. This teaches incorrect evaluation techniques.
claude-sonnet-5-5: pass 2026-10-09 — The factorization, the cancellation, the limit 5/2 at x=-3, and the infinite discontinuity at x=1 are all correct and match the stated answer. Line 5 is a slightly terse way to show the numerator is nonzero, but it is not wrong.
gpt-oss:20b: fail (misleading) 2026-10-09 — The solution incorrectly substitutes the numerator at x=1, writing Subs(-10, x, 1) = -10, whereas the numerator is -10·1−30 = −40. The conclusion that the discontinuity at x=1 is infinite is correct, but the erroneous substitution misleads a student about the calculation.
qwen3.6:27b-mlx: pass 2026-10-09

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 2026-10-09 with SymPy 1.14.0.