Estimating a limit from a table
Problem 1.726 · easy
Make a table of values of \( \displaystyle f(x) = \frac{x^{2} - 5 x + 6}{x^{2} + 4 x - 12} \) at \( \displaystyle x = 2 \pm 0.1,\ 2 \pm 0.01,\ 2 \pm 0.001,\ 2 \pm 0.0001 \) and use it to estimate \( \displaystyle \lim_{x \to 2} f(x) \).
- f(2) itself is undefined (0/0), so the table approaches 2 from both sides.
- The table: f(2 − 0.1) ≈ -1.39240506e-1; f(2 − 0.01) ≈ -1.2640801e-1; f(2 − 0.001) ≈ -1.25140643e-1; f(2 − 0.0001) ≈ -1.25014063e-1; f(2 + 0.0001) ≈ -1.24985938e-1; f(2 + 0.001) ≈ -1.24859393e-1; f(2 + 0.01) ≈ -1.23595506e-1; f(2 + 0.1) ≈ -1.11111111e-1.
- From both sides the values settle toward -0.125000.
- \[ \lim_{x \to 2}\left(\frac{x^{2} - 5 x + 6}{x^{2} + 4 x - 12}\right) = - \frac{1}{8} \]The exact (two-sided) limit agrees with the table.✓ Proved
Answer \( \lim_{x\to 2} f(x) = - \frac{1}{8} \approx -0.125000 \)
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: passqwen3.6:27b-mlx: fail (error) — The table values are incorrect; for example, f(1.9) is approximately -0.135, not -0.139. The solution likely used the wrong function or calculation, leading to a numerically inaccurate table despite the correct final limit.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The table values are incorrect; for example, f(1.9) is approximately -0.135, not -0.139. The solution likely used the wrong function or calculation, leading to a numerically inaccurate table despite the correct final limit.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the indeterminate form, provides accurate numerical values that converge to the stated limit, and confirms the result with an exact calculation.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-18 — The table values listed in step 2 are numerically incorrect (e.g., f(1.9) is approximately -0.139, not -0.13924... wait, actually f(1.9) = -0.001/0.07 = -0.01428... The listed values are roughly 10x too large or correspond to a different function. Specifically, f(1.9) = (3.61 - 9.5 + 6)/(3.61 + 7.6 - 12) = 0.11 / -0.79 ≈ -0.139. The solution lists -0.13924... which is close but let's check f(1.99). f(1.99) = (1.99^2 - 5(1.99) + 6) / (1.99^2 + 4(1.99) - 12). Numerator: 3.9601 - 9.95 + 6 = 0.0101. Denominator: 3.9601 + 7.96 - 12 = -0.0799. Ratio: -0.1264. The solution lists -0.1264... wait, -1.264e-1 is -0.1264. Let's re-read carefully. f(1.9) = 0.11 / -0.79 = -0.13924. Okay, the values are actually correct. My initial quick check was wrong. Let's re-verify f(1.9). 1.9^2 = 3.61. 5*1.9 = 9.5. 3.61 - 9.5 + 6 = 0.11. 1.9^2 + 4*1.9 - 12 = 3.61 + 7.6 - 12 = 11.21 - 12 = -0.79. 0.11 / -0.79 = -0.1392405... The solution says -1.39240506e-1 which is -0.13924... This is correct. Let's check f(1.99). Num: 1.99^2 - 5(1.99) + 6 = 3.9601 - 9.95 + 6 = 0.0101. Denom: 3.9601 + 7.96 - 12 = -0.0799. 0.0101 / -0.0799 = -0.126408... Solution says -1.2640801e-1. Correct. Let's check f(2.1). Num: 4.41 - 10.5 + 6 = -0.09. Denom: 4.41 + 8.4 - 12 = 0.81. -0.09/0.81 = -1/9 = -0.1111... Solution says -1.11111111e-1. Correct. The values are correct. The limit is indeed -1/8 = -0.125. The conclusion is correct. Why did I think it was wrong? I misread the scientific notation or miscalculated. The solution is sound.gpt-oss:20b: pass 2026-10-18
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.