Estimating a limit from a table
Problem 1.591 · easy
Make a table of values of \( \displaystyle f(x) = \frac{\sqrt{x} - 3}{x - 9} \) at \( \displaystyle x = 9 \pm 0.1,\ 9 \pm 0.01,\ 9 \pm 0.001,\ 9 \pm 0.0001 \) and use it to estimate \( \displaystyle \lim_{x \to 9} f(x) \).
- f(9) itself is undefined (0/0), so the table approaches 9 from both sides.
- The table: f(9 − 0.1) ≈ 1.6713222e-1; f(9 − 0.01) ≈ 1.66712989e-1; f(9 − 0.001) ≈ 1.66671297e-1; f(9 − 0.0001) ≈ 1.6666713e-1; f(9 + 0.0001) ≈ 1.66666204e-1; f(9 + 0.001) ≈ 1.66662037e-1; f(9 + 0.01) ≈ 1.66620396e-1; f(9 + 0.1) ≈ 1.66206258e-1.
- From both sides the values settle toward 0.166667.
- \[ \lim_{x \to 9}\left(\frac{\sqrt{x} - 3}{x - 9}\right) = \frac{1}{6} \]The exact (two-sided) limit agrees with the table.✓ Proved
Answer \( \lim_{x\to 9} f(x) = \frac{1}{6} \approx 0.166667 \)
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 (misleading) — The solution lists the right-sided values in increasing order of distance from 9 (0.0001 to 0.1) rather than decreasing order, which contradicts the standard convention for numerical limit tables and obscures the convergence trend. Additionally, the left-sided values are listed in decreasing order of distance, creating an inconsistent presentation.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-17 — The solution lists the right-sided values in increasing order of distance from 9 (0.0001 to 0.1) rather than decreasing order, which contradicts the standard convention for numerical limit tables and obscures the convergence trend. Additionally, the left-sided values are listed in decreasing order of distance, creating an inconsistent presentation.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the indeterminate form, provides accurate numerical values for the specified points, and correctly estimates the limit based on the table.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the indeterminate form, provides accurate numerical values for the specified points, and draws the correct conclusion consistent with the exact limit.gpt-oss:20b: pass 2026-10-14
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.