Estimating a limit from a table
Problem 1.596 · easy
Make a table of values of \( \displaystyle f(x) = \frac{\sin{\left(4 x \right)}}{\sin{\left(x \right)}} \) 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) ≈ 3.90068131; f(0 − 0.01) ≈ 3.99900007; f(0 − 0.001) ≈ 3.99999; f(0 − 0.0001) ≈ 3.9999999; f(0 + 0.0001) ≈ 3.9999999; f(0 + 0.001) ≈ 3.99999; f(0 + 0.01) ≈ 3.99900007; f(0 + 0.1) ≈ 3.90068131.
- From both sides the values settle toward 4.000000.
- \[ \lim_{x \to 0}\left(\frac{\sin{\left(4 x \right)}}{\sin{\left(x \right)}}\right) = 4 \]The exact (two-sided) limit agrees with the table.✓ Proved
Answer \( \lim_{x\to 0} f(x) = 4 \approx 4.000000 \)
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 (style) — [domain objection, downgraded to style] Sentence 1 incorrectly states that the values approach 0 from both sides; they actually approach 4. The statement about f(0) being undefined is correct, but the conclusion about the limit is false.qwen3.6:27b-mlx: pass — The solution correctly identifies the indeterminate form, provides accurate numerical evidence from both sides, and confirms the limit analytically.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the indeterminate form, provides accurate numerical evidence from both sides, and confirms the limit analytically.gpt-oss:20b: fail (style) 2026-10-17 — [domain objection, downgraded to style] Sentence 1 incorrectly states that the values approach 0 from both sides; they actually approach 4. The statement about f(0) being undefined is correct, but the conclusion about the limit is false.qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the indeterminate form, provides accurate numerical estimates consistent with the limit, and confirms the result with symbolic computation.gpt-oss:20b: fail (misleading) 2026-10-14 — Sentence 1 incorrectly states that the values approach 0 from both sides; they actually approach 4. This misleads the reader about the behavior of the function near zero.qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the indeterminate form, provides accurate numerical approximations for the specified points, and correctly infers the limit from the table. The final verification using the exact limit is consistent.gpt-oss:20b: fail (style) 2026-10-14 — [domain objection, downgraded to style] Sentence 1 incorrectly states that the values approach 0 from both sides; they actually approach 4. The statement about f(0) being undefined is correct, but the conclusion about the limit is false.
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 2026-10-17 with SymPy 1.14.0.