Give examples of strictly increasing sequences of ordinals such that

$\displaystyle \text{lim}_n(\alpha_n + \beta) \not = \text{lim}_n \alpha_n + \beta$,

$\displaystyle \text{lim}_n(\alpha_n + \beta_n) \not = \text{lim}_n \alpha_n + \text{lim}_n \beta_n$.

I cannot think any examples that work for this one. The problem is that I can't recall many examples of strictly increasing sequences of ordinals. I need some help on this problem. Thank you.