Let $f(n)$ be the number of prime factors of the positive integer $n$. Find $\lim_{n\to \infty}\frac{f(n)} n$ 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.
Note that $2^{f(n)}\le n$, so $f(n)\le \log_2(n)$. Now the standard tools, such as L'Hospital's Rule, can be used to show that $\lim_{x\to\infty}\frac{\log_2(x)}{x}=0$.
Since we know infinite that there are infinitely many primes, we can take $n$ tending to be an infinitely large prime.

$\dfrac{f(n)}{n}=\dfrac{1}{p}$, here $f(p)=1$(Why?)