Home›Calculus 1›Estimating a limit from a table›Problem 1.655

Estimating a limit from a table

Problem 1.655 · easy

Make a table of values of \( \displaystyle f(x) = \frac{x^{2} + x - 6}{x^{2} + 2 x - 8} \) 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.
Reviewed
The table: f(2 − 0.1) ≈ 8.30508475e-1; f(2 − 0.01) ≈ 8.33055092e-1; f(2 − 0.001) ≈ 8.33305551e-1; f(2 − 0.0001) ≈ 8.33330556e-1; f(2 + 0.0001) ≈ 8.33336111e-1; f(2 + 0.001) ≈ 8.33361106e-1; f(2 + 0.01) ≈ 8.33610649e-1; f(2 + 0.1) ≈ 8.36065574e-1.
Reviewed
From both sides the values settle toward 0.833333.
Reviewed
\[ \lim_{x \to 2}\left(\frac{x^{2} + x - 6}{x^{2} + 2 x - 8}\right) = \frac{5}{6} \]
The exact (two-sided) limit agrees with the table.✓ Proved

Answer \( \lim_{x\to 2} f(x) = \frac{5}{6} \approx 0.833333 \)

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
3	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the indeterminate form, provides accurate numerical approximations that converge to the stated limit, and confirms the result with an exact calculation.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the indeterminate form, provides accurate numerical values converging to the limit, and confirms the result with the exact analytical 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 2026-10-18 with SymPy 1.14.0.