Let $f(n)$ be the number of prime factors of the positive integer $n$. Find $\lim_{n\to \infty}\frac{f(n)} n$

Question

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.

Answer 1

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 $\lim_{x\to\infty}\frac{\log_2(x)}{x}=0$.

Answer 2

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?)

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Answer 3

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 $0$.