Limit of \( \displaystyle \frac{1 - \cos{\left(3 x \right)}}{3 x} \) as \( x \to 0 \)

Problem 1.322 · medium

1. \[ \lim_{x \to 0^+}\left(\frac{1 - \cos{\left(3 x \right)}}{3 x}\right) \]
   limitStart with the original limit.✓ Proved
2. \[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(1 - \cos{\left(3 x \right)}\right)}{\frac{d}{d x} 3 x}\right) \]
   lhopitalApply L'Hopital's rule because the limit is an indeterminate form 0/0.✓ Proved
3. \[ = \lim_{x \to 0^+} \sin{\left(3 x \right)} \]
   simplify simplifyCompute the derivatives. Simplify the expression by canceling the constant 3.✓ Proved
4. \[ = 0 \]
   limitEvaluate the final limit.✓ Proved