Estimating a limit from a table
Problem 1.790 · easy
Make a table of values of \( \displaystyle f(x) = \frac{x^{2} - 2 x}{x^{2} + 6 x} \) at \( \displaystyle x = 0 \pm 0.1,\ 0 \pm 0.01,\ 0 \pm 0.001,\ 0 \pm 0.0001 \) and use it to estimate \( \displaystyle \lim_{x \to 0} f(x) \).
- f(0) itself is undefined (0/0), so the table approaches 0 from both sides.
- The table: f(0 − 0.1) ≈ -3.55932203e-1; f(0 − 0.01) ≈ -3.35559265e-1; f(0 − 0.001) ≈ -3.33555593e-1; f(0 − 0.0001) ≈ -3.33355556e-1; f(0 + 0.0001) ≈ -3.33311111e-1; f(0 + 0.001) ≈ -3.33111148e-1; f(0 + 0.01) ≈ -3.31114809e-1; f(0 + 0.1) ≈ -3.1147541e-1.
- From both sides the values settle toward -0.333333.
- \[ \lim_{x \to 0}\left(\frac{x^{2} - 2 x}{x^{2} + 6 x}\right) = - \frac{1}{3} \]The exact (two-sided) limit agrees with the table.✓ Proved
Answer \( \lim_{x\to 0} f(x) = - \frac{1}{3} \approx -0.333333 \)
Lines: 1 proved, 3 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. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | Not checked | — | a sentence; read, not computed |
| 3 | Not checked | — | a sentence; read, not computed |
| 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 evaluated at a ± 10⁻⁹ in 30-digit arithmetic |
Reviewers
gpt-oss:20b: fail (misleading) — Sentence 1 incorrectly states that the table approaches 0 from both sides; it should say the values approach -1/3.qwen3.6:27b-mlx: pass — The solution correctly identifies the indeterminate form, provides accurate numerical approximations that converge to the stated limit, and confirms the result with an exact symbolic calculation.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-31 — The solution correctly identifies the indeterminate form, provides accurate numerical approximations that converge to the stated limit, and confirms the result with an exact symbolic calculation.gpt-oss:20b: fail (misleading) 2026-10-31 — Sentence 1 incorrectly states that the table approaches 0 from both sides; it should say the values approach -1/3.qwen3.6:27b-mlx: pass 2026-10-30 — The solution correctly identifies the undefined nature of f(0), provides accurate numerical values approaching the limit from both sides, and confirms the result with the exact analytical limit.gpt-oss:20b: fail (misleading) 2026-10-30 — Sentence 1 incorrectly states that the table approaches 0 from both sides; it should say the values approach -1/3, not 0.qwen3.6:27b-mlx: fail (error) 2026-10-30 — The table values are incorrect; for example, f(0.1) is approximately -0.311, not -0.31147541 as listed (the solution appears to have swapped digits or calculated incorrectly). Additionally, the order of values for positive x is reversed in the list, though the conclusion happens to be correct.gpt-oss:20b: fail (misleading) 2026-10-30 — Sentence 1 incorrectly states that the table approaches 0 from both sides; the function actually approaches –1/3, not 0.
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/limit_table, checked 2026-10-31 with SymPy 1.14.0.