Home›Calculus 1›Continuity and types of discontinuity›Problem 1.481

Continuity and types of discontinuity

Problem 1.481 · hard

Find every point where \( \displaystyle f(x) = \frac{12 x + 48}{x^{2} + 5 x + 4} \) 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.
\[ x^{2} + 5 x + 4 = \left(x + 1\right) \left(x + 4\right) \]
Factor the denominator.✓ Proved
\[ \frac{12 x + 48}{x^{2} + 5 x + 4} = \frac{12}{x + 1} \]
The factor (x + 4) cancels.✓ Proved
\[ \lim_{x \to -4^+}\left(\frac{12 x + 48}{x^{2} + 5 x + 4}\right) = -4 \]
At x = -4 the limit exists, but f(-4) is undefined: a removable discontinuity.✓ Proved
\[ \left. 12 \right|_{\substack{ x=-1 }} = 12 \]
At x = -1 the numerator is not 0 while the denominator is, so |f(x)| grows without bound: an infinite discontinuity.✓ Proved

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

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: {-4: 'removable', -1: 'infinite'}

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to check the two-sided limit at x = -4, which is required to classify the discontinuity as removable. Additionally, the reasoning for the infinite discontinuity at x = -1 is flawed because it relies on the simplified expression 12/(x+1) rather than the original function, and it does not verify that the limit is infinite (only that the numerator is non-zero).

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to check the two-sided limit at x = -4, which is required to classify the discontinuity as removable. Additionally, the reasoning for the infinite discontinuity at x = -1 is flawed because it relies on the simplified expression 12/(x+1) rather than the original function, and it does not verify that the limit is infinite (only that the numerator is non-zero).
gpt-oss:20b: pass 2026-10-09
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.