Home›Calculus 1›Estimating a limit from a table›Problem 1.665

Estimating a limit from a table

Problem 1.665 · easy

Make a table of values of \( \displaystyle f(x) = \frac{\tan{\left(2 x \right)}}{x} \) at \( \displaystyle x = 0 \pm 0.1,\ 0 \pm 0.01,\ 0 \pm 0.001,\ 0 \pm 0.0001 \) and use it to estimate \( \displaystyle \lim_{x \to 0} f(x) \).

f(0) itself is undefined (0/0), so the table approaches 0 from both sides.
The table: f(0 − 0.1) ≈ 2.02710036; f(0 − 0.01) ≈ 2.00026671; f(0 − 0.001) ≈ 2.00000267; f(0 − 0.0001) ≈ 2.00000003; f(0 + 0.0001) ≈ 2.00000003; f(0 + 0.001) ≈ 2.00000267; f(0 + 0.01) ≈ 2.00026671; f(0 + 0.1) ≈ 2.02710036.
From both sides the values settle toward 2.000000.
\[ \lim_{x \to 0}\left(\frac{\tan{\left(2 x \right)}}{x}\right) = 2 \]
The exact (two-sided) limit agrees with the table.✓ Proved

Answer \( \lim_{x\to 0} f(x) = 2 \approx 2.000000 \)

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: fail (error) — Sentence 1 incorrectly states that the function values approach 0; they actually approach 2. The statement about approaching 0 from both sides is false and would mislead a student.
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18
gpt-oss:20b: fail (error) 2026-10-18 — Sentence 1 incorrectly states that the function values approach 0; they actually approach 2. The statement about approaching 0 from both sides is false and would mislead a student.
qwen3.6:27b-mlx: pass 2026-10-16
gpt-oss:20b: fail (error) 2026-10-16 — Sentence 1 incorrectly states that the values approach 0 from both sides; the correct behavior is that f(x) approaches 2 as x→0.
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the indeterminate form, provides accurate numerical values for the specified points, and correctly concludes the limit is 2.
gpt-oss:20b: fail (misleading) 2026-10-15 — Sentence 1 incorrectly states that the values approach 0; they actually approach 2. The statement about the table approaching 0 from both sides is false and would mislead a student.

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.