When to use lim-notation and when to use f(x)->L, x->a notation

Question

I was wondering if there is an important distinction between the following two notations for limits:

$a_n\to L$ when $n\to\infty$

and

$\lim_{n\to\infty}a_n=L$

From what I could find it seems like quibbling. I'm currently working with limits of sequences so it's mostly in regard to limits of sequences that I'm asking, however, I am wondering the same thing about limits of function.

Answer 1

They are identical. When you write $\lim_{n \rightarrow \infty} f(n)$, context should be enough to figure out whether $n$ is an integer or a real number. Personally however, I use the arrows for sequences, and the $\lim$ for functions.