Given two sequences of positive numbers $\{a_k\}$ and $\{b_k\} \  (k \in \mathbb N)$ such that:

[b](i)[/b] $a_k < b_k,$

[b](ii) [/b] $\cos a_kx + \cos b_kx \geq -\frac 1k $ for all $k \in \mathbb N$ and $x \in \mathbb  R,$

prove the existence of $\lim_{k \to \infty} \frac{a_k}{b_k}$ and find this limit.