Estimating a limit from a table
Problem 1.724 · easy
Make a table of values of \( \displaystyle f(x) = \frac{x^{2} + x}{x^{2} + 5 x} \) 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) ≈ 1.83673469e-1; f(0 − 0.01) ≈ 1.98396794e-1; f(0 − 0.001) ≈ 1.99839968e-1; f(0 − 0.0001) ≈ 1.99984e-1; f(0 + 0.0001) ≈ 2.00016e-1; f(0 + 0.001) ≈ 2.00159968e-1; f(0 + 0.01) ≈ 2.01596806e-1; f(0 + 0.1) ≈ 2.15686275e-1.
- From both sides the values settle toward 0.200000.
- \[ \lim_{x \to 0}\left(\frac{x^{2} + x}{x^{2} + 5 x}\right) = \frac{1}{5} \]The exact (two-sided) limit agrees with the table.✓ Proved
Answer \( \lim_{x\to 0} f(x) = \frac{1}{5} \approx 0.200000 \)
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 0.2.qwen3.6:27b-mlx: pass — The solution correctly identifies the indeterminate form, provides accurate numerical values approaching the limit from both sides, and correctly concludes the limit is 1/5.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the indeterminate form, provides accurate numerical values approaching the limit from both sides, and correctly concludes the limit is 1/5.gpt-oss:20b: fail (misleading) 2026-10-19 — Sentence 1 incorrectly states that the table approaches 0 from both sides; it should say the values approach 0.2.qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the indeterminate form, provides accurate numerical approximations that converge to the stated limit, and confirms the result with the exact symbolic limit.gpt-oss:20b: fail (misleading) 2026-10-18 — Sentence 1 incorrectly states that the values approach 0; they actually approach 1/5. The conclusion that the table approaches 0 from both sides is false and would mislead a student.qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the indeterminate form, provides accurate numerical approximations that converge to the correct limit, and confirms the result with the exact analytical limit.gpt-oss:20b: fail (misleading) 2026-10-18 — Sentence 1 incorrectly states that the values approach 0 from both sides; they actually approach 1/5. The rest of the solution is fine.
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 2026-10-19 with SymPy 1.14.0.