Solution
= Solution
If $\|x_n^{-1}\|\|x-x_n\|<1$, then
$$
x=x_n\bigl(1+x_n^{-1}(x-x_n)\bigr)
$$
is invertible by the <Neumann series>, a contradiction. Thus every $n$ satisfies
$$
\|x_n^{-1}\|\ge\frac1{\|x-x_n\|}.
$$
Here $x\ne x_n$ because $x_n$ is invertible and $x$ is not. Since $x_n\to x$, this proves the <inverse norm divergence at noninvertible boundary points>:
$$
\boxed{\|x_n^{-1}\|\longrightarrow\infty.}
$$