Home›Calculus 1›Estimating a limit from a table›Problem 1.652

Estimating a limit from a table

Problem 1.652 · easy

Make a table of values of \( \displaystyle f(x) = \frac{x^{2} - 2 x}{x^{2} + 3 x} \) 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) ≈ -7.24137931e-1; f(0 − 0.01) ≈ -6.72240803e-1; f(0 − 0.001) ≈ -6.67222407e-1; f(0 − 0.0001) ≈ -6.66722224e-1; f(0 + 0.0001) ≈ -6.66611113e-1; f(0 + 0.001) ≈ -6.66111296e-1; f(0 + 0.01) ≈ -6.61129568e-1; f(0 + 0.1) ≈ -6.12903226e-1.
From both sides the values settle toward -0.666667.
\[ \lim_{x \to 0}\left(\frac{x^{2} - 2 x}{x^{2} + 3 x}\right) = - \frac{2}{3} \]
The exact (two-sided) limit agrees with the table.✓ Proved

Answer \( \lim_{x\to 0} f(x) = - \frac{2}{3} \approx -0.666667 \)

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 table approaches 0 from both sides; it actually approaches -2/3.
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18
gpt-oss:20b: fail (misleading) 2026-10-18 — Sentence 1 incorrectly states that the table approaches 0 from both sides; it actually approaches -2/3.
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the undefined nature of f(0), provides accurate numerical approximations that converge to the stated limit, and confirms the result with an exact symbolic calculation.
gpt-oss:20b: fail (misleading) 2026-10-16 — Sentence 1 incorrectly states that the table approaches 0 from both sides; it actually approaches -2/3. The rest of the solution is correct.
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The table values are incorrect. For example, f(0.1) = -0.1/0.4 = -0.25, but the table lists -0.6129. The solution likely evaluated the function at x=1 instead of x=0.1 (e.g., f(1) = -1/4 = -0.25 is also not -0.6129, but f(0.1) is definitely not -0.6129; actually f(0.1) = -0.25. Let's re-calculate: f(0.1) = (0.01 - 0.2)/(0.01 + 0.3) = -0.19/0.31 ≈ -0.6129. Ah, the calculation in the table is correct for x=0.1. Wait, let me re-read the problem. x = 0 +/- 0.1. So x=0.1. f(0.1) = (0.1^2 - 2(0.1))/(0.1^2 + 3(0.1)) = (0.01 - 0.2)/(0.01 + 0.3) = -0.19/0.31 ≈ -0.6129. The table value -6.12903226e-1 is -0.6129. This is correct. Let me check x=0.01. f(0.01) = (0.0001 - 0.02)/(0.0001 + 0.03) = -0.0199/0.0301 ≈ -0.6611. The table has -6.61129568e-1. This is correct. Let me check x=0.001. f(0.001) = (0.000001 - 0.002)/(0.000001 + 0.003) = -0.001999/0.003001 ≈ -0.66611. The table has -6.66111296e-1. This is correct. The limit is indeed -2/3. Why did I think it was wrong? Let me check the negative side. x = -0.1. f(-0.1) = (0.01 + 0.2)/(0.01 - 0.3) = 0.21 / -0.29 ≈ -0.7241. The table has -7.24137931e-1. This is correct. The solution seems numerically correct. Is there a conceptual error? The problem asks to estimate the limit. The table shows values approaching -0.666667. The exact limit is -2/3. The solution is sound. Wait, let me look closer at the prompt's instruction. 'Review worked calculus solutions adversarially'. Is there a subtle issue? The function is (x^2 - 2x)/(x^2 + 3x) = x(x-2)/(x(x+3)) = (x-2)/(x+3) for x!=0. Limit as x->0 is -2/3. The table values are correct. The conclusion is correct. Why would it fail? Maybe the scientific notation is considered 'style'? Or maybe I missed a calculation error? Let's check x=0.0001. f(0.0001) = (1e-8 - 0.0002)/(1e-8 + 0.0003) ≈ -0.0002/0.0003 = -2/3. Table: -6.66611113e-1. Correct. x=-0.0001. f(-0.0001) = (1e-8 + 0.0002)/(1e-8 - 0.0003) ≈ 0.0002/-0.0003 = -2/3. Table: -6.66722224e-1. Correct. The solution is actually correct. I will mark it as pass.
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 -2/3.

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.