Limit of \( \displaystyle \frac{\sin{\left(10 x + 5 \right)}}{\sin{\left(4 x + 2 \right)}} \) as \( x \to - \frac{1}{2} \)

Problem 1.316 · medium

1. \[ \lim_{x \to - \frac{1}{2}^+}\left(\frac{\sin{\left(10 x + 5 \right)}}{\sin{\left(4 x + 2 \right)}}\right) \]
   limit factorSet up the limit. Factor the arguments of the sine functions.✓ Proved
2. \[ = \lim_{u \to 0^+}\left(\frac{\sin{\left(5 u \right)}}{\sin{\left(2 u \right)}}\right) \]
   substitutionLet u = 2*x + 1, so u approaches 0 as x approaches -1/2.✓ Proved
3. \[ = \lim_{u \to 0^+}\left(\frac{\frac{d}{d u} \sin{\left(5 u \right)}}{\frac{d}{d u} \sin{\left(2 u \right)}}\right) \]
   lhopitalApply L'Hôpital's rule since the limit is an indeterminate form 0/0.✓ Proved
4. \[ = \lim_{u \to 0^+}\left(\frac{5 \cos{\left(5 u \right)}}{2 \cos{\left(2 u \right)}}\right) \]
   simplifyCompute the derivatives.✓ Proved
5. \[ = \frac{5}{2} \]
   limit simplifyEvaluate the limit by substituting u = 0. Simplify the final value.✓ Proved