# 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 infinty so does the complex numbers, but what exactly does that mean?

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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.