  The first discrete Painlev\'e equation (dPI), which appears in a model of
quantum gravity, is an integrable nonlinear nonautonomous difference equation
which yields the well known first Painlev\'e equation (PI) in a continuum
limit. The asymptotic study of its solutions as the discrete time-step
$n\to\infty$ is important both for physical application and for checking the
accuracy of its role as a numerical discretization of PI. Here we show that the
asymptotic analysis carried out by Boutroux (1913) for PI as its independent
variable approaches infinity can also be achieved for dPI as its discrete
independent variable approaches the same limit.
