TITLE: Let $f(n)$ be the number of prime factors of the positive integer $n$. Find $\lim_{n\to \infty}\frac{f(n)} n$
QUESTION [5 upvotes]: Let $f(n)$ be the number of prime factors of the positive integer $n$. Find $\displaystyle \lim_{n\to \infty}\frac{f(n)} n$.

I suspect it's equal to $0$, but how can I show this? Thank you.

REPLY [3 votes]: The integer $n$ can have at most one prime factor exceeding $\sqrt n$; therefore $f(n) \le \sqrt n + 1$, which quickly implies that the limit equals $0$.