Limits of complex numbers > "We say $z_n \rightarrow \infty$ if, for each positive number $M$ (no matter how large), there is an integer $N$ such that $|z_n |>M$ whenever $n > N$; similary $\lim_{z\rightarrow z_0}f(z) =\infty$ means that for each positive number $M$, there is a $\delta >0$ such that $|f(z)|>M$ whenever $0<|z-z_0|<\delta$." I don't understand what this means? I think it says when the magnitude goes to infinity so does the complex numbers, but what exactly does that mean?

The precise definitions are given in what you wrote. If you want a more intuitive explanation, the first clause states that a sequence $\\{z_n\\}$ approaches $\infty$ if given any $M$, there is some index $N$ in the sequence such that every term after that point in the sequence is at least distance $M$ away from the origin in the complex plane.

The second clause explains that the notation $\lim_{z\to z_0}f(z)=\infty$ means that for any $M$, there exists a $\delta>0$ such that if $z$ is a point contained within the circle (besides possibly $z_0$) of radius $\delta$ around $z_0$, then the image $f(z)$ of $z$ is at least distance $M$ away from the origin.