Home›Calculus 1›Continuity and types of discontinuity›Problem 1.288

Continuity and types of discontinuity

Problem 1.288 · hard

Find every point where \( \displaystyle f(x) = \frac{3 x - 6}{x^{2} - 7 x + 10} \) 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.
\[ x^{2} - 7 x + 10 = \left(x - 5\right) \left(x - 2\right) \]
Factor the denominator.✓ Proved
\[ \frac{3 x - 6}{x^{2} - 7 x + 10} = \frac{3}{x - 5} \]
The factor (x − 2) cancels.✓ Proved
\[ \lim_{x \to 2^+}\left(\frac{3 x - 6}{x^{2} - 7 x + 10}\right) = -1 \]
At x = 2 the limit exists, but f(2) is undefined: a removable discontinuity.✓ Proved
\[ \left. 3 \right|_{\substack{ x=5 }} = 3 \]
At x = 5 the numerator is not 0 while the denominator is, so |f(x)| grows without bound: an infinite discontinuity.✓ Proved

Answer \( \text{removable at } x = 2;\ \text{infinite at } x = 5 \)

Lines: 4 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in 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: {2: 'removable', 5: 'infinite'}

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — Step 5 incorrectly substitutes x=5 into the simplified numerator '3' to check for an infinite discontinuity, rather than checking the original numerator '3x-6' (which is 9, not 0). While the final classification is correct, the reasoning presented is mathematically flawed and misleading.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 5 incorrectly substitutes x=5 into the simplified numerator '3' to check for an infinite discontinuity, rather than checking the original numerator '3x-6' (which is 9, not 0). While the final classification is correct, the reasoning presented is mathematically flawed and misleading.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 5 incorrectly substitutes x=5 into the simplified numerator 3 instead of the original numerator (3x-6), which equals 9 at x=5. While the classification of the discontinuity as infinite is correct, the reasoning contains a factual arithmetic error.
gpt-oss:20b: fail (error) 2026-10-05 — The solution incorrectly evaluates the numerator at x=5 as 3 instead of 9, yet still concludes an infinite discontinuity. The numeric mistake is a factual error.

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.