The limit of $z\cdot\overline{z}$ as $z\to i$ How can I compute this limit:
$$\lim_{z\rightarrow i} (z\cdot \overline{z})$$
(where $\overline{z}$ is the conjugate of $z$)?

As $\mathbb{C}$ comes nicely equipped with a metric, limits behave as nice as ever. Therefore, the limit of the product is the product of the limits. And the limit of the conjugate is the conjugate of the limit (this can be proved using a standard epsilon-delta argument). Hence, in your example:
$$\lim_{z \to i} (z \cdot \overline{z})=\lim_{z \to i}(z) \lim_{z \to i}(\overline{z})= i \cdot(-i) = 1.$$
