8. Find a solution to the functional equation $f^{(3)}(x)=\frac{x}{1+x}, x \geqslant 0$.

Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly.
8. Let $f(x)=\frac{a x+b}{c x+d}$, then $f(f(x))=\frac{a f(x)+b}{c f(x)+d}=\frac{\left(a^{2}+b x\right) x+a b+b d}{(a c+c d) x+b x+d^{2}}$, $f^{(3)}(x)=\frac{\left(a^{3}+2 a b c+b d\right) x+b\left(a^{2}+b c+a d+d^{2}\right)}{\left(a^{2} c+b x^{2}+a c d+c d^{2}\right) x+a b c+2 b c d+d^{3}}$, let $b=0$, then $f^{(0)}(x)=\frac{a^{3} x}{\left(a^{2} c+a c d+c d^{2}\right) x+d^{3}}$, thus we have $a^{2} c+a c d+c d^{2}=a^{3}, d^{3}=a^{3}$, solving these gives $a=d=3 c$. Specifically, we can take $a=3, b=0, c=1, d=3$, $f(x)=\frac{3 x}{x+3}$.