sequences and series - Does the limit of $f(x) = sumlimits_{n=1}^{+infty} frac{(-1)^n sin frac{x}{n}}{n}$ exist?

Consider the function $f: [0; +\infty] \rightarrow \mathbb{R}$ given by the formula $$f(x) = \sum\limits_{n=1}^{+\infty} \frac{(-1)^n \sin \frac{x}{n}}{n}$$
Does $\lim\limits_{x\rightarrow 0} f(x)$ exist? If yes, find it's value.
Is $f$ differantiable? If yes, check if $f'(0) > 0$.

Any ideas how to do that? I assume this is about functions series, it's pointwise convergent I think, does that mean that this limit exists?

Thanks a lot for your help!