Home›Calculus 1›Estimating a limit from a table›Problem 1.669

Estimating a limit from a table

Problem 1.669 · easy

Make a table of values of \( \displaystyle f(x) = \frac{x^{2} - 9}{x^{2} + 4 x - 21} \) at \( \displaystyle x = 3 \pm 0.1,\ 3 \pm 0.01,\ 3 \pm 0.001,\ 3 \pm 0.0001 \) and use it to estimate \( \displaystyle \lim_{x \to 3} f(x) \).

f(3) itself is undefined (0/0), so the table approaches 3 from both sides.
Reviewed
The table: f(3 − 0.1) ≈ 5.95959596e-1; f(3 − 0.01) ≈ 5.995996e-1; f(3 − 0.001) ≈ 5.99959996e-1; f(3 − 0.0001) ≈ 5.99996e-1; f(3 + 0.0001) ≈ 6.00004e-1; f(3 + 0.001) ≈ 6.00039996e-1; f(3 + 0.01) ≈ 6.003996e-1; f(3 + 0.1) ≈ 6.03960396e-1.
Reviewed
From both sides the values settle toward 0.600000.
Reviewed
\[ \lim_{x \to 3}\left(\frac{x^{2} - 9}{x^{2} + 4 x - 21}\right) = \frac{3}{5} \]
The exact (two-sided) limit agrees with the table.✓ Proved

Answer \( \lim_{x\to 3} f(x) = \frac{3}{5} \approx 0.600000 \)

✓ 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 values converging to the limit, and confirms the result with an exact calculation.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the indeterminate form, provides accurate numerical values converging to the limit, and confirms the result with an exact calculation.
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 consistent with the limit, and confirms the result with the exact analytical limit.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the indeterminate form, provides accurate numerical approximations consistent with the limit, and confirms the result with an exact calculation.
gpt-oss:20b: pass 2026-10-15

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.