Let me try to expand a little bit the problem (so it's too long for a usual comment).  

Consider the determinant $D_N=D_N(\lambda;a_1,\dots,a_{N-1};b_1,\dots,c_{N-1})$ of the corresponding matrix $\lambda-A$. Expanding the determinant along the first row gives
$$
D_N(\lambda;a_1,\dots,a_{N-1};b_1,\dots,b_{N-1})
=\lambda D_{N-1}(\lambda;a_2,\dots,a_{N-1};b_2,\dots,b_{N-1})
-a_1b_1D_{N-2}(\lambda;a_3,\dots,a_{N-1};b_3,\dots,b_{N-1});
$$
in other words,
$$
D_N/D_{N-1}=\lambda-\frac{a_1b_1}{D_{N-1}/D_{N-2}}
=\dots
=\lambda-\frac{a_1b_1}{\lambda-\dfrac{a_2b_2}{\lambda-\dfrac{a_3b_3}{\dots
-\dfrac{a_{N-1}b_{N-1}}{\lambda}}}}.
$$
In order to get some information about the asymptotics of the zero(s) of $D_N(\lambda)/D_{N-1}(\lambda)$ one really have to have some knowledge about the $a_ib_i$, $i=1,2,\dots$. This reduces the problem to a problem for the related family of orthogonal polynomials and even Deift's book is too advanced, it is the best source on this.