You're saying that if $n$ is very large, $x_n$ is very large, and so $\frac1{x_n}$ tends to $0$,
But then you said that if $n$ is very large (tends to infinity), then $x_{n+1}$ is very close to $2$. This also means that $x_n$ is very close to $2$. But that contradicts your earlier statement about $x_n$ being very large.