limits - does $lim_{Ntoinfty}frac{sum_{i=1}^N a_i}{sum_{i=1}^N b_i}$ converge to $lim_{Ntoinfty}frac{1}{N}sum_{i=1}^Nfrac{a_i}{b_i}$

Can this ever be the case?
$$\lim\limits_{N\to\infty}\frac{\sum\limits_{i=1}^N a_i}{\sum\limits_{i=1}^N b_i} = \lim\limits_{N\to\infty}\frac{1}{N}\sum_{i=1}^N\frac{a_i}{b_i}$$ with $a_i>0$, $b_i>0$, $a_iothers pointed out simulations indicate convergence, but is there formal ground to it?