Limit of \( \displaystyle \frac{\sin{\left(15 x + 5 \right)}}{\sin{\left(6 x + 2 \right)}} \) as \( x \to - \frac{1}{3} \)

Problem 1.320 · medium

1. \[ \lim_{x \to - \frac{1}{3}^+}\left(\frac{\sin{\left(15 x + 5 \right)}}{\sin{\left(6 x + 2 \right)}}\right) \]
   limitIdentify the limit and check for indeterminate form.✓ Proved
2. \[ = \lim_{x \to - \frac{1}{3}^+}\left(\frac{\frac{d}{d x} \sin{\left(15 x + 5 \right)}}{\frac{d}{d x} \sin{\left(6 x + 2 \right)}}\right) \]
   lhopitalApply L'Hopital's rule because the limit is of the form 0/0.✓ Proved
3. \[ = \lim_{x \to - \frac{1}{3}^+}\left(\frac{5 \cos{\left(15 x + 5 \right)}}{2 \cos{\left(6 x + 2 \right)}}\right) \]
   simplifyCompute the derivatives of the numerator and denominator.✓ Proved
4. \[ = \lim_{x \to - \frac{1}{3}^+} \frac{5}{2} \]
   substitution simplifySubstitute x = -1/3 into the expression. Evaluate the trigonometric functions.✓ Proved
5. \[ = \frac{5}{2} \]
   simplifySimplify the final fraction.✓ Proved