\begin{eqnarray*}L &=& \lim_{x\rightarrow 0}{3x-\sin 3x\over 27x^3}\\ &=& \lim_{x\rightarrow 0}{3x-3\sin x+4\sin^3x\over 27x^3}\\ &=& \lim_{x\rightarrow 0}\left({x-\sin x\over 9x^3}+{4\over 27}\left({\sin x\over x}\right)^3\right)\\ &=& {1\over 9}L+{4\over 27}\end{eqnarray*}