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Vertical and horizontal asymptotes

Problem 1.475 · medium

Find all vertical and horizontal asymptotes of \( \displaystyle f(x) = \frac{- 3 x^{3} + 12 x^{2} - 3 x + 12}{x^{2} + 2 x - 8} \).

\[ x^{2} + 2 x - 8 = \left(x - 2\right) \left(x + 4\right) \]
Factor the denominator.✓ Proved
The numerator is not zero at x = -4 or x = 2, so both are vertical asymptotes.
The numerator has higher degree than the denominator, so f(x) grows without bound as x → ±∞: there is no horizontal asymptote.

Answer \( x = -4,\ x = 2;\ \text{no horizontal asymptote} \)

Lines: 1 proved, 2 not checked. 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	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Not checked	claude-sonnet-5-5 script, run by sympy 1.14.0	claude-sonnet-5-5 script, run by sympy 1.14.0: script failed: SyntaxError: invalid syntax (`checks/1.475/line2-b013f4ea.py`)
3	Not checked	claude-sonnet-5-5 script, run by sympy 1.14.0	claude-sonnet-5-5 script, run by sympy 1.14.0: script failed: SyntaxError: invalid syntax (`checks/1.475/line3-cf0bfc70.py`)
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	f probed just beside each vertical asymptote and at x = ±1e9

Reviewers

gpt-oss:20b: fail (misleading) — The solution incorrectly states that there is no horizontal asymptote. Since the numerator has degree 3 and the denominator degree 2, the function has an oblique (slant) asymptote, not a horizontal one. The conclusion about horizontal asymptotes is false and would mislead a student.
qwen3.6:27b-mlx: pass — The solution correctly identifies the vertical asymptotes by checking that the numerator is non-zero at the roots of the denominator. It also correctly concludes there is no horizontal asymptote because the degree of the numerator exceeds the degree of the denominator.

Senior review claude-sonnet-5-5, 2026-10-09: pass — The numerator is nonzero at x = 2 (value 30) and x = -4 (value 408), so both vertical asymptotes are valid. Because the numerator's degree exceeds the denominator's, f is unbounded as x → ±∞, so there is no horizontal asymptote.

gpt-oss:20b: dismiss — The numerator's degree is one higher than the denominator's, so there is a slant asymptote y = -3x + 18, but the problem asks only for vertical and horizontal asymptotes, and 'no horizontal asymptote' is correct.

Every verdict on record (5)

gpt-oss:20b: fail (misleading) 2026-10-09 — The solution incorrectly states that there is no horizontal asymptote. Since the numerator has degree 3 and the denominator degree 2, the function has an oblique (slant) asymptote, not a horizontal one. The conclusion about horizontal asymptotes is false and would mislead a student.
qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly identifies the vertical asymptotes by checking that the numerator is non-zero at the roots of the denominator. It also correctly concludes there is no horizontal asymptote because the degree of the numerator exceeds the degree of the denominator.
claude-sonnet-5-5: pass 2026-10-09 — The numerator is nonzero at x = 2 (value 30) and x = -4 (value 408), so both vertical asymptotes are valid. Because the numerator's degree exceeds the denominator's, f is unbounded as x → ±∞, so there is no horizontal asymptote.
gpt-oss:20b: fail (error) 2026-10-09 — The function has a slant (oblique) asymptote because the numerator’s degree exceeds the denominator’s by one. The solution incorrectly states that there is no horizontal asymptote.
qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly identifies the vertical asymptotes by checking that the denominator is zero while the numerator is non-zero at those points. It also correctly applies the degree comparison rule to conclude there is no horizontal asymptote.

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/asymptotes, checked 2026-10-09 with SymPy 1.14.0.