Limit of \( \displaystyle \frac{\sin{\left(5 x \right)}}{\sin{\left(2 x \right)}} \) as \( x \to 0 \)

Problem 1.16 · medium

1. \[ \lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{\sin{\left(2 x \right)}}\right) \]
   limitIdentify the limit to be evaluated.✓ Proved
2. \[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(5 x \right)}}{\frac{d}{d x} \sin{\left(2 x \right)}}\right) \]
   lhopitalThe limit is an indeterminate form 0/0, so apply L'Hôpital's rule.✓ Proved
3. \[ = \lim_{x \to 0^+}\left(\frac{5 \cos{\left(5 x \right)}}{2 \cos{\left(2 x \right)}}\right) \]
   simplifyCompute the derivatives of the numerator and denominator.✓ Proved
4. \[ = \frac{5}{2} \]
   limit simplifyEvaluate the limit by substituting x = 0. Simplify the resulting numerical expression.✓ Proved