Limit of \( \displaystyle \frac{2 x}{\sin{\left(2 x \right)}} \) as \( x \to 0 \)
Problem 1.6 · easy Video
Evaluate \( \displaystyle \lim_{x \to 0} \frac{2 x}{\sin{\left(2 x \right)}} \).
- \[ \lim_{x \to 0^+}\left(\frac{2 x}{\sin{\left(2 x \right)}}\right) \]limitStart with the original limit expression.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right) \]simplifySimplify the fraction by dividing numerator and denominator by 2.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \sin{\left(x \right)}}\right) \]lhopitalApply L'Hôpital's Rule because the limit is in the indeterminate form 0/0.✓ Proved
- \[ = \lim_{x \to 0^+} \frac{1}{\cos{\left(x \right)}} \]simplifyCompute the derivatives of the numerator and denominator.✓ Proved
- \[ = 1 \]limit simplifyEvaluate the limit by substituting x = 0. Calculate the final value.✓ Proved
Answer \( 1 \)
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. A reviewer's objection to the explanation stands; see the receipt. How we verify
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 undefined where sin(2*x) = 0 undefined where sin(x) = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x) = 0 undefined where Derivative(sin(x), x) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Derivative(sin(x), x) = 0 undefined where cos(x) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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: fail (error) — Step 2 incorrectly simplifies 2*x/sin(2*x) to x/sin(x). The correct simplification uses sin(2x)=2 sin(x) cos(x), yielding x/(sin(x) cos(x)), not x/sin(x).qwen3.6:27b-mlx: fail (error) — Step 2 claims to simplify 2*x/sin(2*x) to x/sin(x) by dividing numerator and denominator by 2, which is algebraically incorrect (sin(2*x)/2 is not sin(x)). This is a mathematical error, not just a labeling issue.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 2 claims to simplify 2*x/sin(2*x) to x/sin(x) by dividing numerator and denominator by 2, which is algebraically incorrect (sin(2*x)/2 is not sin(x)). This is a mathematical error, not just a labeling issue.gpt-oss:20b: fail (error) 2026-10-03 — Step 2 incorrectly simplifies 2*x/sin(2*x) to x/sin(x). The correct simplification uses sin(2x)=2 sin(x) cos(x), yielding x/(sin(x) cos(x)), not x/sin(x).qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 2 claims to simplify 2*x/sin(2*x) to x/sin(x) by dividing numerator and denominator by 2, which is algebraically incorrect (the denominator would become sin(2*x)/2, not sin(x)). This is a mathematical error, not just a labeling issue.gpt-oss:20b: fail (error) 2026-09-29 — Step 2 incorrectly simplifies 2*x/sin(2*x) to x/sin(x). The correct simplification uses sin(2x)=2sin(x)cos(x), yielding x/(sin(x)cos(x)), not x/sin(x).qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 2 incorrectly transforms the limit expression. Dividing the numerator and denominator by 2 yields x/sin(2x), not x/sin(x). The solver effectively changed the argument of the sine function without justification, which is mathematically invalid.gpt-oss:20b: fail (error) 2026-09-29 — Step 2 incorrectly simplifies 2*x/sin(2*x) to x/sin(x). The correct simplification would involve sin(2x)=2sin(x)cos(x), yielding x/(sin(x)cos(x)), not x/sin(x).
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.