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Vertical and horizontal asymptotes

Problem 1.184 · medium

Find all vertical and horizontal asymptotes of \( \displaystyle f(x) = \frac{- 4 x - 16}{x^{2} - 1} \).

\[ x^{2} - 1 = \left(x - 1\right) \left(x + 1\right) \]
Factor the denominator.✓ Proved
The numerator is not zero at x = -1 or x = 1, so both are vertical asymptotes.
Reviewed
\[ \lim_{x \to \infty}\left(\frac{- 4 x - 16}{x^{2} - 1}\right) = 0 \]
Divide through by the highest power of x in the denominator.✓ Proved
\[ \lim_{x \to -\infty}\left(\frac{- 4 x - 16}{x^{2} - 1}\right) = 0 \]
The same as x → −∞.✓ Proved

Answer \( x = -1,\ x = 1;\ y = 0 \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ 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	f probed just beside each vertical asymptote and at x = ±1e9

Reviewers

gpt-oss:20b: pass
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 determines the horizontal asymptote by evaluating limits at infinity.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies vertical asymptotes by checking that the denominator is zero while the numerator is non-zero. It also correctly determines the horizontal asymptote by evaluating limits at infinity.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the vertical asymptotes by checking that the denominator is zero while the numerator is non-zero. It also correctly identifies the horizontal asymptote by evaluating the limits at infinity.
gpt-oss:20b: pass 2026-10-04

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 2026-10-04 with SymPy 1.14.0.