Writing a limit so that the subscript goes directly underneath

I have used the following which puts x -> y under the limit.

\lim\limits_{x \to y}

One way would be to use \scalebox from the graphicx package to rescale the version obtained by \displaystyle. Here is a comparison of the usual inline version of \lim and the scaled version:

\documentclass{article}
\usepackage{xcolor}
\usepackage{graphicx}

\newcommand{\Text}{\noindent Lorem ipsum dolor sit amet.  
Lorem ipsum dolor sit amet. 
Lorem ipsum dolor sit amet }

\newcommand{\Lim}[1]{\raisebox{0.5ex}{\scalebox{0.8}{$\displaystyle \lim_{#1}\;$}}}

\usepackage{graphicx}
\begin{document}
\Text 
$\textcolor{blue}{\lim_{x \rightarrow p} f(x) = q}$
\Text
$\textcolor{red}{\Lim{x \rightarrow p} f(x) = q}$
\Text 
\end{document}

I'm not sure if this is squished enough for you but the amsmath package allows you to put something under something else through the \underset command:

\documentclass[11pt]{article}
\usepackage{amsmath}
\pagestyle{empty}
\begin{document}
$\underset{x \to \infty}{\lim}$
\end{document}