Home›Calculus 1›Estimating a limit from a table›Problem 1.727

Estimating a limit from a table

Problem 1.727 · easy

Make a table of values of \( \displaystyle f(x) = \frac{x^{2} - x - 2}{x^{2} + 5 x + 4} \) 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) ≈ -1.06896552; f(-1 − 0.01) ≈ -1.00668896; f(-1 − 0.001) ≈ -1.00066689; f(-1 − 0.0001) ≈ -1.00006667; f(-1 + 0.0001) ≈ -9.99933336e-1; f(-1 + 0.001) ≈ -9.99333555e-1; f(-1 + 0.01) ≈ -9.93355482e-1; f(-1 + 0.1) ≈ -9.35483871e-1.
Reviewed
From both sides the values settle toward -1.000000.
Reviewed
\[ \lim_{x \to -1}\left(\frac{x^{2} - x - 2}{x^{2} + 5 x + 4}\right) = -1 \]
The exact (two-sided) limit agrees with the table.✓ Proved

Answer \( \lim_{x\to -1} f(x) = -1 \approx -1.000000 \)

✓ 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 for the specified points, and correctly concludes the limit based on the table.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the indeterminate form, provides accurate numerical approximations for the specified points, and correctly concludes the limit based on the table.
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 approaching the limit from both sides, and correctly concludes the limit is -1.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-18
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.