Limit of sum of factions of finite constants and matrix

Question

Suppose $\mathbf{a}_{i}$ is a $2\times 1$ constant vector with $\frac{1}{N}% \sum_{i=1}^{N}\mathbf{a}_{i}\mathbf{a}_{i}^{T}$ converges to a finite $% 2\times 2$ matrix as $N\rightarrow \infty $, and $b_{i}$ is a finite positive constant with $\frac{1}{N}\sum_{i=1}^{N}b_{i}$ converges to a finite constant as $% N\rightarrow \infty $, then what is the limit of the following form \begin{equation} \frac{1}{N}\sum_{i=1}^{N}\frac{\mathbf{a}_{i}\mathbf{a}_{i}^{T}}{\mathbf{a}% _{i}^{T}\mathbf{a}_{i}+b_{i}}, \end{equation} as $N\rightarrow \infty ,$ does it have a close form? Thanks