Reversing the digits of an infinite decimal

The limit exists precisely when the sequence of digits is eventually constant. If it is eventually constantly $d$, the limit is $\frac{d}9$.

Suppose that $\lim_{k \to \infty} x^R_k = x^R$ exists. Then, for $k$ large enough, the first digit of $x^R_k$ is equal to the first digit of $x^R$, say $a$.

Hence, $d_k = d_{k+1} = \dots = a$ and $x^R = 0.aaaa \dots = a/9$.

So, the number $x$ must have a decimal expansion where eventually only the digit $a$ appears.