On the behaviour of $\left(1+\frac{1}{n+1}\right)^{n+1}-\left(1+\frac{1}{n}\right)^n$ I have to find the limit : (let $k\in \mathbb{R}$)

$$\lim_{n\to \infty}n^k \left(\Big(1+\frac{1}{n+1}\Big)^{n+1}-\Big(1+\frac{1}{n}\Big)^n \right)=?$$

My Try :
$$\lim_{n\to \infty}\frac{n^k}{\Big(1+\frac{1}{n}\Big)^n} \left(\frac{\Big(1+\frac{1}{n+1}\Big)^{n+1}}{\Big(1+\frac{1}{n}\Big)^n}-1\right)$$
we know that :
$$\frac{\Big(1+\frac{1}{n+1}\Big)^{n+1}}{\Big(1+\frac{1}{n}\Big)^n}>1$$
now what do i do ?

$$\lim_{n\to \infty}n^k \left((1+\frac{1}{n+1})^{n+1}-(1+\frac{1}{n})^n \right)=
\lim_{n\to \infty}n^k \left(\frac{e}{2n^2}+O((\frac{1}{n^3})) \right)$$
for n<2 limit is 0, for n=2 limit is e/2, for n>2 limit is infinity
