> If you write a limit as \lim_{n \to \infty} etc etc then LaTex gets to decide whether to write your limit in inline style (next to the limit) or below. If you want it always to be below then you need to write
In display mode, without modifications, LaTeX has no choice; `$$\lim_{n \to \infty}$$` in the default LaTeX compiler will always put `{n \to \infty}` below `\lim`, unless you or some package have monkeyed with things to change that default behavior. (In inline mode, similarly, $\lim_{n \to \infty}$ will by default always place `{n \to \infty}` as a subscript of `\lim`, unless you or some package have changed the default behavior.)
If you write a limit as \lim_{n \to \infty} etc etc then LaTex gets to decide whether to write your limit in inline style (next to the limit) or below. If you want it always to be below then you need to write
\lim\limits_{n \to \infty}
It’s the same deal with summations and integrals.