  The limit $q$-Durrmeyer operator, $D_{\infty,q},$ was introduced and its
approximation properties were investigated by V. Gupta in 2008 during a study
of $q$-analogues for the Bernstein-Durrmeyer operator. In the present work,
this operator is investigated from a different perspective. More precisely, the
growth estimates are derived for the entire functions comprising the range of
$D_{\infty,q}$. The interrelation between the analytic properties of a function
$f$ and the rate of growth for $D_{\infty,q}f$ are established, and the
sharpness of the obtained results are demonstrated.
