calculus - Prove the limit is $sqrt{e}$.

How do you show

$$\lim\limits_{k \rightarrow \infty} \frac{\left(2+\frac{1}{k}\right)^k}{2^k}=\sqrt{e}$$

I know that

$$\lim\limits_{k \to \infty} \left(1+\frac{1}{k}\right)^k=e$$ but I don't know how to apply this.

Answer

Hint:

$$\frac{a^k}{b^k}=\left(\frac ab\right)^k,$$ and in general, $$\lim_{k\to\infty}\left(1+\frac xk\right)^k=e^x.$$