Calculate the following series limit: $\lim_{n \to \infty} (\frac{n}{n+3})^\sqrt{n(1+n)}$ Calculate the following series limit: $$\lim_{n \to \infty} \left(\frac{n}{n+3}\right)^\sqrt{n(1+n)}$$
I'm struggling with this limit problem. I changed over to this form: $e^{\sqrt{n(1+n)}\ln(\frac{n}{n+3})}$ but I'm not sure how to continue from here, tried using Lhopital but it just ended up being nasty maybe I used it too early?

It is:
$$\lim_{n \to \infty} \left(\frac{n}{n+3}\right)^\sqrt{n(1+n)}=\lim_{n \to \infty} \left(1+\frac{3}{n}\right)^{-\sqrt{n(1+n)}}=\lim_{n \to \infty} \left[\left(1+\frac{3}{n}\right)^{\frac{n}{3}}\right]^{-\frac{3\sqrt{n(1+n)}}{n}}=e^{-3}.$$
