Compute the following limit: $$\Large \lim_{n\to\infty} n\log \left(\frac{n+1}{n}\right ).$$
My solution.
$$\lim_{n\to\infty} n\log \left(\frac{n+1}{n}\right)=\lim_{n\to\infty}\log\left(\frac{n+1}{n}\right)^n=\lim_{n\to\infty}\log\left(1+\frac{1}{n}\right)^n\\\color{red}{=}\log\left(\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n\right)\\=\log(e)\\=1$$
In the red equality I used the fact that $\log(n)$ is continuous in $n=e$.
Is my solution correct? Are there any other ways to determine this limit?