Definition

We say that L \in \mathbb{C} is the limit of a Complex Functions (f : D \to \mathbb{C}), f at z_{0} \in \mathbb{C} if the following holds:

For every \varepsilon \gt 0 there exists \delta \gt 0 such that, if \mid z - z_{0} \mid \; < \delta then \mid f(z) - L \mid \; \lt \varepsilon. (Then we write L = \lim_{ z \to z_{0} } f(z))

Observe that this uses the absolute value of complex numbers.