Find $\displaystyle\lim_{n\to\infty} a_n$ $$a_n:=\left(\frac{1}{k}+\frac{1}{n}\right)^n,k\in\mathbb N$$
My work:
I write it as $$\lim_{n\to\infty}{\left[ \frac{1}{k} \cdot k \cdot \left(\frac{1}{k}+\frac{1}{n}\right) \right]^n} =\lim_{n\to\infty}{\left(\frac{1}{k}\left(1+\frac{k}{n}\right)\right)^n}=\lim_{n\to\infty}{\left(\frac{1}{k}\right)^n}\cdot\lim_{n\to\infty}{\left(1+\frac{k}{n}\right)^n}=0\cdot e^k=0$$
Can someone tell me whether my attempt is on the right track?