Limit of \( \displaystyle \frac{3 x}{\sin{\left(3 x \right)}} \) as \( x \to 0 \)

Problem 1.69 · easy

1. \[ \lim_{x \to 0^+}\left(\frac{3 x}{\sin{\left(3 x \right)}}\right) \]
   limitIdentify the limit to evaluate.✓ Proved
2. \[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 3 x}{\frac{d}{d x} \sin{\left(3 x \right)}}\right) \]
   lhopitalApply L'Hopital's rule since the limit is an indeterminate form 0/0.✓ Proved
3. \[ = \lim_{x \to 0^+} \frac{1}{\cos{\left(3 x \right)}} \]
   simplify simplifyCompute the derivatives. Simplify the fraction by canceling the common factor 3.✓ Proved
4. \[ = 1 \]
   limit simplify simplifyEvaluate the limit by substituting x = 0. Evaluate the cosine function. Final result.✓ Proved