Home›Calculus 1›Estimating a limit from a table›Problem 1.725

Estimating a limit from a table

Problem 1.725 · easy

Make a table of values of \( \displaystyle f(x) = \frac{x^{2} + 2 x - 3}{x^{2} + 5 x - 6} \) at \( \displaystyle x = 1 \pm 0.1,\ 1 \pm 0.01,\ 1 \pm 0.001,\ 1 \pm 0.0001 \) and use it to estimate \( \displaystyle \lim_{x \to 1} f(x) \).

f(1) itself is undefined (0/0), so the table approaches 1 from both sides.
Reviewed
The table: f(1 − 0.1) ≈ 5.65217391e-1; f(1 − 0.01) ≈ 5.70815451e-1; f(1 − 0.001) ≈ 5.71367338e-1; f(1 − 0.0001) ≈ 5.71422449e-1; f(1 + 0.0001) ≈ 5.71434694e-1; f(1 + 0.001) ≈ 5.71489787e-1; f(1 + 0.01) ≈ 5.72039943e-1; f(1 + 0.1) ≈ 5.77464789e-1.
Reviewed
From both sides the values settle toward 0.571429.
Reviewed
\[ \lim_{x \to 1}\left(\frac{x^{2} + 2 x - 3}{x^{2} + 5 x - 6}\right) = \frac{4}{7} \]
The exact (two-sided) limit agrees with the table.✓ Proved

Answer \( \lim_{x\to 1} f(x) = \frac{4}{7} \approx 0.571429 \)

✓ 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 — The solution correctly identifies the indeterminate form, provides accurate numerical approximations consistent with the limit, and confirms the result with the exact analytical limit.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the indeterminate form, provides accurate numerical approximations consistent with the limit, and confirms the result with the exact analytical limit.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the indeterminate form, provides accurate numerical values for the specified points, and correctly estimates the limit consistent with the exact analytical result.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the indeterminate form, provides accurate numerical values for the specified points, and correctly estimates the limit based on the table. The final exact limit confirms the numerical result.
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.