Home›Calculus 1›Estimating a limit from a table›Problem 1.594

Estimating a limit from a table

Problem 1.594 · easy

Make a table of values of \( \displaystyle f(x) = \frac{2^{x} - 1}{x} \) 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) ≈ 6.69670085e-1; f(0 − 0.01) ≈ 6.90750456e-1; f(0 − 0.001) ≈ 6.9290701e-1; f(0 − 0.0001) ≈ 6.93123158e-1; f(0 + 0.0001) ≈ 6.93171204e-1; f(0 + 0.001) ≈ 6.93387463e-1; f(0 + 0.01) ≈ 6.95555006e-1; f(0 + 0.1) ≈ 7.17734625e-1.
From both sides the values settle toward 0.693147.
\[ \lim_{x \to 0}\left(\frac{2^{x} - 1}{x}\right) = \ln{\left(2 \right)} \]
The exact (two-sided) limit agrees with the table.✓ Proved

Answer \( \lim_{x\to 0} f(x) = \ln{\left(2 \right)} \approx 0.693147 \)

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 (misleading) — Sentence 1 incorrectly states that the values approach 0 from both sides; they actually approach ln 2. The description of the limit’s behavior is false.
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17
gpt-oss:20b: fail (misleading) 2026-10-17 — Sentence 1 incorrectly states that the values approach 0 from both sides; they actually approach ln 2. The description of the limit’s behavior is false.
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the indeterminate form, provides accurate numerical values for the specified points, and correctly estimates the limit based on the trend in the table.
gpt-oss:20b: fail (misleading) 2026-10-14 — Sentence 1 incorrectly states that the table approaches 0 from both sides; it should say the values approach ln(2) ≈0.693147.
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the indeterminate form, provides accurate numerical approximations for the specified points, and correctly concludes the limit based on the trend. The final verification with the exact limit is consistent.
gpt-oss:20b: fail (misleading) 2026-10-14 — Sentence 1 incorrectly states that the values approach 0; they actually approach ln 2. The description of the limit’s behavior is misleading.

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 2026-10-17 with SymPy 1.14.0.