Say I have $f(x)$ defined as
$$f(x) = \begin{cases} 40 & x \geq 10000 \\ -3x & x < 10000\end{cases}$$
Here I understood rather easily that $\lim_{x\to +\infty} f(x) = 40$.
But now I have this:
$$f(x) = \begin{cases} -x & x\in\mathbb{Q} \\\\ 2 & x\in\mathbb{R}\backslash\mathbb{Q} \end{cases}$$
How to get, if existent, the limit for $x\to+\infty$?
SO I thought: when I go to infinity, I can go though "rational steps" or "non rational steps". Either ways, I can reach infinity so a priori I cannot exclude any of the two options.
So the limit doesn't exist, and I thought it to be for some similar reason of when we study what happens at the infinity for $(-1)^n$. If it's "an even infinity" then limit is $1$, whereas it's $0$ for an odd infinity.
Am I right or wrong?
Thank you!