find a limit depending on parametres $k,l\in \mathbb N$

Question

Let $ k,l\in \mathbb {N}$, find $$ \lim_{n\to\infty} \frac{(n+1)^k + (-n)^l}{(n-1)^k -n^l}.$$

I am really stuck, I do not know where to start. I tried to find different case when k is bigger than l or when k is smaller, but it seems it is getting me nowhere.

Thanks for help

Answer 1

If $k>l$ divide numerator and denominator by $n^k$: $$ \frac{(n+1)^k + (-n)^l}{(n-1)^k -n^l}=\frac{(1+1/n)^k+(-1)^ln^{l-k}}{(1-1/n)^k+n^{l-k}} $$ I leave to you the cases $k<l$ and $k=l$.