Estimating a limit from a table
Problem 1.662 · easy
Make a table of values of \( \displaystyle f(x) = \frac{x^{2} + x}{x^{2} + 4 x} \) 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) ≈ 2.30769231e-1; f(0 − 0.01) ≈ 2.48120301e-1; f(0 − 0.001) ≈ 2.49812453e-1; f(0 − 0.0001) ≈ 2.4998125e-1; f(0 + 0.0001) ≈ 2.5001875e-1; f(0 + 0.001) ≈ 2.50187453e-1; f(0 + 0.01) ≈ 2.51870324e-1; f(0 + 0.1) ≈ 2.68292683e-1.
- From both sides the values settle toward 0.250000.
- \[ \lim_{x \to 0}\left(\frac{x^{2} + x}{x^{2} + 4 x}\right) = \frac{1}{4} \]The exact (two-sided) limit agrees with the table.✓ Proved
Answer \( \lim_{x\to 0} f(x) = \frac{1}{4} \approx 0.250000 \)
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 function approaches 0 from both sides; it should say the function approaches 1/4. This could mislead a student about the limit’s value.qwen3.6:27b-mlx: pass — The solution correctly identifies the indeterminate form, provides accurate table values converging to 1/4, and confirms the limit analytically.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the indeterminate form, provides accurate table values converging to 1/4, and confirms the limit analytically.gpt-oss:20b: fail (misleading) 2026-10-18 — Sentence 1 incorrectly states that the function approaches 0 from both sides; it should say the function approaches 1/4. This could mislead a student about the limit’s value.qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the indeterminate form, provides accurate table values that converge to the limit, and confirms the result with symbolic calculation.gpt-oss:20b: fail (misleading) 2026-10-16 — Sentence 1 incorrectly states that the table approaches 0 from both sides; it actually approaches 1/4. This misleads a student about the behavior of the function near 0.qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the undefined point, provides accurate numerical values for the specified points, and correctly infers the limit from the table. The final check confirms the result.gpt-oss:20b: fail (misleading) 2026-10-15 — Sentence 1 incorrectly states that the table approaches 0 from both sides; the values actually approach 0.25, 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 2026-10-18 with SymPy 1.14.0.