In the well-known coupon collector problem, let $X$ be the expected time for a collector to obtain all different $n$ coupons. It is well known that
$E[X]=n\log n +O(n)$
where $O(n)$ is a term linear in $n$.
I want to write that this must be true in expectation, say
Statement 1: $\lim_{n\to \infty} E[X]=n\log n$
So far, I only have seen that
$\frac{X}{n \log n} \to 1$
Would my statement 1 be correct? Is there any other way to say that $E(X)$ converges to $n \log n$? Why everybody writes convergence as a ratio, and not $X \to n \log n$? (The arrow represents convergence in probability)
The important part is to understand that your statement 1 is not right or wrong, it makes no sense, as on the left you have a limit over $n$, while on the right you have a term that still contains $n$.
What is your statement 1 is supposed to mean? Arguebly, that $E[X]$ and $n\log{}n$ are 'near each other' when n increases to infinity. But since both values got to infinity, different person's understanding what it means to be 'near each other' might differ.
One could say that their quotient should tend to 1, which is what you wrote down and is correct in this case. Another might say that their difference should tend to 0, which is not correct in this case.
As written, statement 1 makes not sense. The various interpretations are partly true and partly not, which is the reason nobody tries to extend the definiton of 'limit' to make your statement 1 contain some meaning. That meaning can already be expressed in different terms, and you gave 2 of those meanings yourself.