Vertical and horizontal asymptotes
Problem 1.469 · medium
Find all vertical and horizontal asymptotes of \( \displaystyle f(x) = \frac{- 4 x^{3} + 4 x^{2} - 4 x + 4}{x^{2} - 9} \).
- \[ x^{2} - 9 = \left(x - 3\right) \left(x + 3\right) \]Factor the denominator.✓ Proved
- The numerator is not zero at x = -3 or x = 3, 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 \( x = -3,\ x = 3;\ \text{no horizontal asymptote} \)
Lines: 1 proved, 2 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | a sentence; read, not computed |
| 3 | Not checked | — | a sentence; read, not computed |
| 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 (error) — The solution incorrectly states that there is no horizontal asymptote. Since the numerator’s degree (3) exceeds the denominator’s degree (2) by one, the function has an oblique (slant) asymptote, not none.qwen3.6:27b-mlx: pass — The solution correctly identifies vertical asymptotes by checking that the denominator is zero while the numerator is non-zero. It also correctly applies the degree comparison rule to conclude there is no horizontal asymptote.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-09 — The solution incorrectly states that there is no horizontal asymptote. Since the numerator’s degree (3) exceeds the denominator’s degree (2) by one, the function has an oblique (slant) asymptote, not none.qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly identifies vertical asymptotes by checking that the denominator is zero while the numerator is non-zero. It also correctly applies the degree comparison rule to conclude there is no horizontal asymptote.gpt-oss:20b: pass 2026-10-09qwen3.6:27b-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.