Is it possible to write such a limit notation respect to $n$?

Question

How can I write this notation correctly?

I want to write,

For example,

$$\lim_{n\to\infty} f(n)=A$$

But, here $n$ must be $n=4k, k\in\mathbb{Z^{+}}$.

$f (n) $ is such a function that, for $f (n)$ the limit can not be calculated if $n≠4k, k\in\mathbb{Z^{+}}$

$f(n)$ is so complicated function. I want to write limit notation only respect to $n$. I know possible notation:

$$\lim_{k\to\infty} f(4k)=A$$

How can I express this feature in the limit formula? Is there such a notation?

Answer 1

One option: $$\lim_{{n\to \infty}\\{\ \ 4\mid n}}f(n)$$

Answer 2

I think that $\displaystyle\lim_{k\to\infty}f(4k)=A$ is the best option, but you can also use $\displaystyle\lim_{\begin{array}{c}k\to\infty\\k\in4\mathbb{Z}\end{array}}f(k)=A$.

Answer 3

I would write

$$\lim\limits_{\substack{n \to \infty \\ n \in 4\mathbb Z}} f(n) = A$$