  For $N \geq 2$, we study a certain sequence $(\rho_p^{(c_p)})$ of
N-dimensional representations of the mapping class group of the one-holed torus
arising from SO(3)-TQFT, and show that the conjecture of Andersen, Masbaum, and
Ueno \cite{1} holds for these representations. This is done by proving that, in
a certain basis and up to a rescaling, the matrices of these representations
converge as $p$ tends to infinity. Moreover, the limits describe the action of
$SL_{2}(\mathbb{Z})$ on the space of homogeneous polynomials of two variables
of total degree $N-1$.
