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

Problem 1.150 · medium

1. \[ \lim_{x \to \frac{1}{2}^+}\left(\frac{\sin^{2}{\left(2 x - 1 \right)}}{\left(2 x - 1\right)^{2}}\right) \]
   limit algebra substitutionStart with the original limit. Rewrite the square of the fraction. Let u = 2*x - 1.✓ Proved
2. \[ = \lim_{u \to 0^+}\left(\frac{\sin^{2}{\left(u \right)}}{u^{2}}\right) \]
   substitutionAs x approaches 1/2, u approaches 0.✓ Proved
3. \[ = \lim_{u \to 0^+}\left(\frac{\frac{d}{d u} \frac{\sin^{2}{\left(u \right)}}{u^{2}}}{\frac{d}{d u} 1}\right) \]
   lhopitalApply L'Hôpital's rule because the limit is of form 0/0.Not checked
4. \[ = \lim_{u \to 0^+}\left(\frac{2 u^{2} \sin{\left(u \right)} \cos{\left(u \right)} - 2 u \sin^{2}{\left(u \right)}}{u^{4}}\right) \]
   simplifyCompute the derivative of the numerator and denominator.Not checked
5. \[ = \lim_{u \to 0^+}\left(\frac{2 u^{2} \sin{\left(2 u \right)} - 2 u \sin^{2}{\left(u \right)}}{u^{4}}\right) \]
   simplifyUse the double angle identity.Not checked
6. \[ = \lim_{u \to 0^+}\left(\frac{2 \sin{\left(2 u \right)} - \frac{2 \sin^{2}{\left(u \right)}}{u}}{u^{2}}\right) \]
   simplifySimplify the expression by dividing by u.Not checked
7. \[ = \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right) \]
   limit-lawThe limit simplifies to the fundamental trigonometric limit.Not checked
8. \[ = \lim_{u \to 0^+}\left(\frac{\frac{d}{d u} \sin{\left(u \right)}}{\frac{d}{d u} u}\right) \]
   lhopitalApply L'Hôpital's rule again.✓ Proved
9. \[ = \lim_{u \to 0^+} \cos{\left(u \right)} \]
   simplifyEvaluate the derivative.✓ Proved
10. \[ = 1 \]
    limitEvaluate the final limit.✓ Proved