There are four styles used in typesetting math formulas which affect the size and certain formatting parameters (notably the placement of sub and superscripts on variable size symbols):
\textstyle: default in the running text and in array environment
\displaystyle: default for displayed equations
\scriptstyle:: default for first-level sub and superscripts
\scriptscriptstyle: default for higher-level sub and superscripts
Take a look at the following document to see how the appearance changes when different styles are used:
\documentclass{article}
\usepackage{amsmath}
\begin{document}
$\textstyle\sum_{i=1}^\infty x_i$,
$\displaystyle\sum_{i=1}^\infty x_i$,
$\scriptstyle\sum_{i=1}^\infty x_i$,
$\scriptscriptstyle\sum_{i=1}^\infty x_i$
\end{document}

Now, coming back to your specific query, use \displaystyle.
\documentclass{article}
\usepackage{amsmath}
\begin{document}
$\varprojlim_n$ $\displaystyle\varprojlim_n$
\end{document}

\varprojlim\limits_{n}– egreg Aug 25 '17 at 08:48