By definition, the limit operator $\mathbb{L}$ in sequences with real terms $\{a_n\}$ commutes.
For example,
$$\color{red}{\lim_n} \color{blue}{\ln n} = \color{blue}{\ln} \color{red}{\lim_n n} \tag 1$$
$$\color{red}{\lim_n} \color{teal}{\sqrt n} = \sqrt{\color{red}{\color{blue}{\lim_n n}}} \tag 2$$
I know that when $f$ is continuous in $c\in (a,b)\subseteq \Bbb R$, ($\displaystyle\lim_{x\to c} f(x)=f(c)=f(\displaystyle\lim_{x\to c} x)$), $f$ and the limit easily commute.
But if I have not introduced the concept of continuity, how can I make it clear that the limit operator commutes, in an appropriate way, with other operators (see logarithm, $n-$th root, etc.)? Is it possible to give this as a postulate, or is it possible to create a simple proof?
I am looking for easier way to show the limit operator commute to fifth-year scientific high school students aged $17-18$.