argument of sequence of complex numbers

Question

How can I find a sequence $z_n = x_n+iy_n \in \mathbb{C}\setminus\{ 0\}$ with its limit $z = x + iy \in \mathbb{C}\setminus\{ 0\}$ such that $arg(z_n) \rightarrow -\pi$ and $arg(z) = \pi$? where $arg(z) = tan^{-1}(\frac{y}{x})$

Thanks for any hint.

Answer 1

Take $$z_n = -1 - \frac{i}{n}\to -1$$ when $n\to\infty$. You can check easily that $\arg z_n\to -\pi$ while $\arg(-1) = \pi$.