Home›Calculus 1›Trigonometric limits›Problem 1.637

Limit of \( \displaystyle \frac{2 x - 1}{\sin{\left(2 x - 1 \right)}} \) as \( x \to \frac{1}{2} \)

Problem 1.637 · medium Video

Evaluate \( \displaystyle \lim_{x \to \frac{1}{2}} \frac{2 x - 1}{\sin{\left(2 x - 1 \right)}} \).

\[ \lim_{x \to \frac{1}{2}^+}\left(\frac{2 x - 1}{\sin{\left(2 x - 1 \right)}}\right) \]
limit algebra algebraEvaluate the limit of the given function. Factor out the 2 in the numerator. Rewrite the numerator to match the argument of the sine function.✓ Proved
\[ = \lim_{u \to 0^+}\left(\frac{2 u}{\sin{\left(2 u \right)}}\right) \]
substitutionLet u = x - 1/2, so as x approaches 1/2, u approaches 0.✓ Proved
\[ = \lim_{u \to 0^+}\left(\frac{\frac{d}{d u} 2 u}{\frac{d}{d u} \sin{\left(2 u \right)}}\right) \]
lhopitalApply L'Hopital's rule since the limit is an indeterminate form 0/0.✓ Proved
\[ = \lim_{u \to 0^+} \frac{1}{\cos{\left(2 u \right)}} \]
simplify simplifyCompute the derivatives of the numerator and denominator. Simplify the fraction.✓ Proved
\[ = 1 \]
limit simplifyEvaluate the limit by substituting u = 0. Final result.✓ Proved

Answer \( 1 \)

The graph of f(x) = (2x - 1)/sin(2x - 1). Two points slide along the curve toward x = 1/2, one from each side; their heights close in on 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. 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	✓ 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 - 1) = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(2*x - 1) = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(2x - 1) = 0 undefined where sin(2u) = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(2u) = 0 undefined where Derivative(sin(2u), u) = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where Derivative(sin(2u), u) = 0 undefined where cos(2u) = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(2*u) = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(2*u) = 0
9	✓ 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: pass
qwen3.6:27b-mlx: fail (error) — Step 5 misapplies L'Hôpital's rule by replacing the limit of a quotient with the limit of the quotient of derivatives, which is not algebraically equal to the previous line. L'Hôpital's rule is a logical equivalence for the limit value, not an algebraic transformation of the expression itself, so the consecutive lines are not equal.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — Step 5 misapplies L'Hôpital's rule by replacing the limit of a quotient with the limit of the quotient of derivatives, which is not algebraically equal to the previous line. L'Hôpital's rule is a logical equivalence for the limit value, not an algebraic transformation of the expression itself, so the consecutive lines are not equal.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — Step 2 violates the 'one change' rule by factoring the numerator and simultaneously expanding the argument of the sine function in the denominator. Step 3 is also defective because it claims to rewrite the numerator to match the denominator, but the numerator in Step 2 (2*(x-1/2)) is already algebraically equal to the denominator's argument (2*x-1), making the step redundant or mislabeled.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies algebraic manipulation, substitution, and L'Hopital's rule. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-10-14

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 2026-10-17 with SymPy 1.14.0.