Simple question on limit of the modulus I know that for real functions, if the limit at infinity of $f$ is a number $l$, then the limit of $|f|$ is $|l|$. Is this true for complex functions too? Thanks.
I'm thinking that if a function vanishes at infinity, then its modulus (distance from origin) also vanishes. And vice-versa. Correct?

Yes, and it is true for every function with values in a normed vector space. This follows from the reverse triangular inequality
$$
|\|f(x)\|-\|l\||\leq\|f(x)-l\|.
$$
In other words, the map $y\to \|y\|$ is $1$-Lipschitz, hence continuous. 
In particular, if $\lim_{+\infty}f(x)$ exists, then $\lim_{+\infty}\|f(x)\|=\|\lim_{+\infty}f(x)\|$.
Note: in $\mathbb{C}$, replace $\|\cdot\|$ by $|\cdot|$.
