Limit of \( \displaystyle \frac{\tan{\left(2 x \right)}}{2 x} \) as \( x \to 0 \)
Problem 1.113 · easy
- \[ \lim_{x \to 0^+}\left(\frac{\tan{\left(2 x \right)}}{2 x}\right) \]limitStart with the given limit.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan{\left(2 x \right)}}{\frac{d}{d x} 2 x}\right) \]lhopitalApply L'Hopital's rule because the limit is an indeterminate form 0/0.✓ Proved
- \[ = \lim_{x \to 0^+} \sec{\left(2 x \right)} \]simplify simplifyCompute the derivatives of the numerator and denominator. Simplify the expression by canceling the constant 2.✓ Proved
- \[ = 1 \]limitEvaluate the limit as x approaches 0.✓ Proved
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.
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 undefined where Derivative(2*x, 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 Derivative(2*x, x) = 0 sec has poles at odd multiples of pi/2 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 sec has poles at odd multiples of pi/2 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 sec has poles at odd multiples of pi/2 |
| 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) — The derivative of tan(2x) is 2·sec²(2x), not 2·sec(2x). The solution applies an incorrect rule in step 2, leading to an incorrect simplification and ultimately a wrong limit evaluation.qwen3.6:27b-mlx: pass — The solution correctly applies L'Hopital's rule and simplifies the resulting expression. Each step changes only one aspect of the problem and uses valid labels from the fixed vocabulary.
Senior review claude-sonnet-5-5, 2026-10-03: fail — Line 3 misstates the derivative of tan(2x) as 2*sec(2x) instead of 2*sec(2x)^2, and line 4 carries the error forward. The final answer 1 is correct, but the intermediate lines are not equal as expressions. Line 3 also uses the label 'simplify' for a differentiation step.
gpt-oss:20b: uphold — d/dx tan(2x) = 2*sec(2x)^2, so line 3's numerator 2*sec(2x) is false as a function. The SymPy 'proof' only compared two limit values, both 1, and not the integrands.
Every verdict on record (5)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies L'Hopital's rule and simplifies the resulting expression. Each step changes only one aspect of the problem and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-10-03 — The derivative of tan(2x) is 2·sec²(2x), not 2·sec(2x). The solution applies an incorrect rule in step 2, leading to an incorrect simplification and ultimately a wrong limit evaluation.claude-sonnet-5-5: fail (error) 2026-10-03 — Line 3 misstates the derivative of tan(2x) as 2*sec(2x) instead of 2*sec(2x)^2, and line 4 carries the error forward. The final answer 1 is correct, but the intermediate lines are not equal as expressions. Line 3 also uses the label 'simplify' for a differentiation step.qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies L'Hopital's rule to the 0/0 indeterminate form, computes the derivatives accurately, and simplifies the resulting expression to find the limit.gpt-oss:20b: fail (error) 2026-10-03 — Step 3 incorrectly differentiates tan(2*x); the derivative is 2*sec^2(2*x), not 2*sec(2*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.