Let $f:R\longrightarrow R$,and $f(x)=x-\dfrac{x^2}{2}$,and $$L_{1}(x)=f(x),L_{2}(x)=f(f(x)),L_{3}(x)=f(L_{2}(x)),\cdots,L_{n}(x)=f(L_{n-1}(x))$$
and let $$a_{n}=L_{n}\left(\dfrac{17}{n}\right)$$ prove that $$\displaystyle\lim_{n\to\infty}\left(na_{n}\right)$$ is exist, and find the value limit.
my idea: I want find this $$L_{n}(x)=\cdots?$$ But it is very ugly, becasuse $$L_{2}(x)=f(f(x))=\left(x-\dfrac{x^2}{2}\right)-\dfrac{(x-\dfrac{x^2}{2})^2}{2}=-\dfrac{1}{8}x^4-\dfrac{1}{2}x^3-x^2+x$$