What is $\lim_{z \to 0} \frac{1}{z}$ in $\mathbb{C}$? I have the limit $$\lim_{z \to 0} \frac{1}{z}$$
that I want to find for $z \in \mathbb{C}$. If for $z = x+iy$ with $x = \operatorname{Re}{z}$ and $y = \operatorname{Im}{z}$, I restrict $z$ to the positive real axis $z = x$, I obtain the limit $$\lim_{x \to 0} \frac{1}{x} = \pm \infty.$$ Since the limit approaches different values, the limit does not exist. Is this correct? This thread
Limit of $f(z)=\frac 1z$ as $z$ approaches $0$? mentions the limit to be $\infty_{\mathbb{C}}$ so I am confused.
 A: In $\mathbb{C}$, there is no distinction between $\infty$ and $- \infty$. This is because, in $\mathbb{C}$, one can approach $\infty$ in any direction. Therefore, use the convention that
$$\lim_{z \to a} f(z) = \infty \iff \lim_{z \to a} |f(z)| = \infty,$$
where the last limit is a real limit and is defined in the real sense. The concept of a point at infinity in $\mathbb{C}$ has a very elegant interpretation through Stereographic projection and the Riemann-sphere. To put it very shortly, we can define what is sometimes called the extended complex plane $\mathbb{C}_{\infty} = \mathbb{C} \cup \{ \infty \}$, which is isometric to the Riemann-sphere (read about it here: https://en.wikipedia.org/wiki/Riemann_sphere). Topoligists will sometimes refer to $\mathbb{C}_{\infty}$ as the one-point compactification of $\mathbb{C}$, meaning that it is a compact metric space differing from $\mathbb{C}$ by just one point.
Edit: Now to the limit in question. You have already observed that the limit does not exist in the sense that it does not approach a complex number. With that in mind, try to evaluate the limit using the equivalence above. We consider
$$\lim_{z \to 0} \left| \frac{1}{z} \right| = \lim_{z \to 0} \frac{1}{|z|} =  \lim_{r \to 0^+} \frac{1}{r}=\infty.$$
