Coefficient of $x^n\text{ in }\frac{x}{(x-1{)}^2(x-2)}\text{ is }{C}_n\text{ then }\underset{n\to \mathrm{\infty }}{lim} \frac{C_n}{n}=$

$\frac{x}{(x-1{)}^2(x-2)}=\frac{-2}{x-1}-\frac{1}{(x-1{)}^2}+\frac{2}{x-2}$

$=2[1-x{]}^{-1}-[1-x{]}^{-2}-{\left[1-\frac{x}{2}\right]}^{-1}$

$=2\left[1+x+x^2+\dots +x^n+\dots \right]-\left[1+2x+\dots +(n+1)x^n\dots ..\right]$

$+\left[1+\frac{x}{2}+\dots \dots .+{\left(\frac{x}{2}\right)}^n+\dots \dots \right]$

$\text{ Coefficient of }{x}^n=C_n$

$=2-(n+1)-\frac{1}{2^n}=1-n-\frac{1}{2^n}$

$L{t}_{n\to \mathrm{\infty }}\frac{C_n}{n}=-1$