Describe the limiting behaviour of the following sequences:

1) $\displaystyle \dfrac{\ln n}{n^a}$, $\displaystyle a>0$

I think its 0. I worked this out because I know $\displaystyle n^a$ grows faster than $\displaystyle \ln n$. Is that good enough to do that in a exam situation, or would you need to prove it.

2) $\displaystyle \dfrac{n\sin (n\pi/4)}{\sqrt{n^2+1}}$

I think this diverges, but I'm not too sure how to show it.

3) $\displaystyle \dfrac{n+\cos n\pi}{n-\cos n\pi}$

Would I divide top and bottom by $\displaystyle n$ and so the sequence would converge to -1