  For $q\in(0,1),$ let $B_q$ denote the limit $q$-Bernstein operator. In this
paper, the distance between $B_q$ and $B_r$ for distinct $q$ and $r$ in the
operator norm on $C[0,1]$ is estimated, and it is proved that $1\leqslant
\|B_q-B_r\|\leqslant 2,$ where both of the equalities can be attained. To
elaborate more, the distance depends on whether or not $r$ and $q$ are rational
powers of each other. For example, if $r^j\neq q^m$ for all $j,m\in
\mathbb{N},$ then $\|B_q-B_r\|=2,$ and if $r=q^m, m\in \mathbb{N},$ then
$\|B_q-B_r\|=2(m-1)/m.$
