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

Problem 1.466 · 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 the 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.✓ Proved
4. \[ = \lim_{x \to \frac{1}{3}^+} \frac{5}{2} \]
   limitEvaluate the limit by substituting x = 1/3.✓ Proved
5. \[ = \frac{5}{2} \]
   algebra simplify simplify simplify simplifySimplify the arguments of the trigonometric functions. Evaluate the arithmetic inside the cosine functions. Substitute the value of cos(0). Simplify the fraction. Final simplified value.✓ Proved