Home›Calculus 1›Continuity and types of discontinuity›Problem 1.757

Continuity and types of discontinuity

Problem 1.757 · hard

Find every point where \( \displaystyle f(x) = \frac{2 x^{2} - 2}{x^{2} - 5 x + 4} \) 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.
Reviewed
\[ x^{2} - 5 x + 4 = \left(x - 4\right) \left(x - 1\right) \]
Factor the denominator.✓ Proved
\[ \frac{2 x^{2} - 2}{x^{2} - 5 x + 4} = \frac{2 x + 2}{x - 4} \]
The factor (x − 1) cancels.✓ Proved
\[ \lim_{x \to 1^+}\left(\frac{2 x^{2} - 2}{x^{2} - 5 x + 4}\right) = - \frac{4}{3} \]
At x = 1 the limit exists, but f(1) is undefined: a removable discontinuity.✓ Proved
\[ \left. 2 x + 2 \right|_{\substack{ x=4 }} = 10 \]
At x = 4 the numerator is not 0 while the denominator is, so |f(x)| grows without bound: an infinite discontinuity.✓ Proved

Answer \( \text{removable at } x = 1;\ \text{infinite at } x = 4 \)

✓ 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. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: {1: 'removable', 4: 'infinite'}

Reviewers

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

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-25
gpt-oss:20b: pass 2026-10-25
qwen3.6:27b-mlx: pass 2026-10-24
gpt-oss:20b: pass 2026-10-24
qwen3.6:27b-mlx: pass 2026-10-24 — The solution correctly identifies the points of discontinuity, factors the rational function, and properly classifies the discontinuities based on the behavior of the simplified function and limits.
gpt-oss:20b: pass 2026-10-24

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.