Given that ${x}_{0}$ is the unique positive solution of $(2-x{)}^{n+1}=x(x+1)\cdots (x+n)$, try to find the asymptotic value of

$M=\prod _{k=0}^{n}{\left(\frac{k+2}{k+{x}_{0}}\right)}^{k+2}$

with absolute error $o(1)$ as $n\to \mathrm{\infty}$, where ${H}_{n}$ denotes n-th harmonic number $\sum _{k=1}^{n}1/k$

$M=\prod _{k=0}^{n}{\left(\frac{k+2}{k+{x}_{0}}\right)}^{k+2}$

with absolute error $o(1)$ as $n\to \mathrm{\infty}$, where ${H}_{n}$ denotes n-th harmonic number $\sum _{k=1}^{n}1/k$