sequences and series - Limit $lim_{ntoinfty}nfrac{sinfrac{1}{n}-frac{1}{n}}{1+frac{1}{n}}$

I need to find a limit of a sequence:
$$\lim_{n\to\infty}n\frac{\sin\frac{1}{n}-\frac{1}{n}}{1+\frac{1}{n}}$$

I tried to divide numerator and denominator by n, but it didn't help, as the limit became $\frac{0}{0}$. I tried other things, but always got an indefinite limit. I know that the limit is 0, but I just don't know how to show it. It's probably something really simple, but I'm totally stuck.