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

Problem 1.155 · 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 algebraStart with the original limit. Apply the power rule for quotients.✓ Proved
2. \[ = \lim_{u \to 0^+}\left(\frac{\sin^{2}{\left(u \right)}}{u^{2}}\right) \]
   substitution algebraSubstitute u = 2*x + 1, which implies u -> 0 as x -> -1/2. Distribute the square to the numerator and denominator.✓ Proved
3. \[ = \lim_{u \to 0^+}\left(\frac{\frac{d}{d u} \sin^{2}{\left(u \right)}}{\frac{d}{d u} u^{2}}\right) \]
   lhopitalApply L'Hopital's rule because the limit is an indeterminate form 0/0.✓ Proved
4. \[ = \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)} \cos{\left(u \right)}}{u}\right) \]
   simplify simplify algebraCompute the derivatives. Cancel the common factor of 2. Split the expression into a product.✓ Proved
5. \[ = \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right) \lim_{u \to 0^+} \cos{\left(u \right)} \]
   limit-lawApply the product rule for limits.✓ Proved
6. \[ = 1 \]
   limit simplify simplifyEvaluate the first limit component. Evaluate the second limit component. Final result.✓ Proved