  An asymptotic expression of the orthonormal polynomials $\mathcal{P}_{N}(z)$
as $N\rightarrow\infty$, associated with the singularly perturbed Laguerre
weight $w_{\alpha}(x;t)=x^{\alpha}{\rm
e}^{-x-\frac{t}{x}},~x\in[0,\infty),~\alpha>-1,~t\geq0$ is derived. Based on
this, we establish the asymptotic behavior of the smallest eigenvalue,
$\lambda_{N}$, of the Hankel matrix generated by the weight $w_{\alpha}(x;t)$.
