Limit of \( \displaystyle \frac{3 x - \sin{\left(3 x \right)}}{27 x^{3}} \) as \( x \to 0 \)

Problem 1.57 · medium

1. \[ \lim_{x \to 0^+}\left(\frac{3 x - \sin{\left(3 x \right)}}{27 x^{3}}\right) \]
   limitSet up the limit with the specified direction.✓ Proved
2. \[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(3 x - \sin{\left(3 x \right)}\right)}{\frac{d}{d x} 27 x^{3}}\right) \]
   lhopitalApply L'Hopital's rule since the limit is an indeterminate form 0/0.✓ Proved
3. \[ = \lim_{x \to 0^+}\left(\frac{3 - 3 \cos{\left(3 x \right)}}{81 x^{2}}\right) \]
   simplifyCompute the derivatives of the numerator and denominator.✓ Proved
4. \[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(3 - 3 \cos{\left(3 x \right)}\right)}{\frac{d}{d x} 81 x^{2}}\right) \]
   lhopitalApply L'Hopital's rule again as the limit is still 0/0.✓ Proved
5. \[ = \lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{18 x}\right) \]
   simplify simplifyCompute the derivatives of the numerator and denominator. Simplify the fraction by dividing the coefficients.✓ Proved
6. \[ = \lim_{x \to 0^+} \frac{1}{6} \]
   trig-limit cancelUse the fundamental trigonometric limit sin(u)/u -> 1 as u -> 0. Cancel the common factor of x.✓ Proved
7. \[ = \frac{1}{6} \]
   limitEvaluate the final constant value.✓ Proved