Limit of \( \displaystyle \frac{\tan{\left(3 x \right)}}{3 x} \) as \( x \to 0 \)
Problem 1.90 · easy
Evaluate \( \displaystyle \lim_{x \to 0} \frac{\tan{\left(3 x \right)}}{3 x} \).
- \[ \lim_{x \to 0^+}\left(\frac{\tan{\left(3 x \right)}}{3 x}\right) \]limitStart with the given limit.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{x}\right) \]substitutionLet u = 3x. As x approaches 0, u also approaches 0.✓ Proved
- \[ = 1 \]trig-limitApply the fundamental trigonometric limit lim_{u->0} tan(u)/u = 1.✓ Proved
Answer \( 1 \)
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where x = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where x = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy took the limit from both sides and got the stated value |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies substitution to transform the limit into a standard form and then evaluates it using the fundamental trigonometric limit. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies substitution to transform the limit into a standard form and then evaluates it using the fundamental trigonometric limit. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-30 — The solution correctly applies substitution to transform the limit into a standard form and then evaluates it using the known trigonometric limit. Each step changes only one aspect of the expression and uses valid labels.gpt-oss:20b: pass 2026-09-30
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
gemma4:26b, checked 2026-10-03 with SymPy 1.14.0.