If $\displaystyle \lim_{\alpha\rightarrow \infty} \frac{\alpha^2 + 1}{\alpha + 1} - a\alpha$ exists, find $\displaystyle a\in \mathbb{R}$ and the limit.

EDIT: It has to be $\displaystyle \alpha\rightarrow \infty$. Nice book I have *Dripping in sarcasm* And I agree with the finite part of course.