Question about limits in lim command and scale in mathtools package

Question

This is a follow up question regarding the limits in \lim command and mathtools package. To begin with how can I make the limits to start at the beggining of the lim symbol and not ahead of it? Meaning the lim symbol and the limits not to be centered.

Also in the mathtool package if I use the scale option then the limits are scaled but as it seems to me around a point and so as the scale becomes smaller the larger the vertical gap between the limits and the lim symbol becomes. How can I fix that so that the vertical space will remail the same as the original in the lim command? And why is that happening?

This is a the code I used to try some cases, which belongs to Peter Grill who posted it to answer a previous question of mine.

\documentclass[12pt]{article}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{calc}

% http://tex.stackexchange.com/questions/60453/reducing-font-size-in-equation/
\newcommand*{\Scale}[2][4]{\scalebox{#1}{$#2$}}%
\newcommand*{\Resize}[2]{\resizebox{#1}{!}{$#2$}}%

\begin{document}
\noindent
Without \verb|\mathclap|:
\[  \lim_{n\to +\infty}x(n) \]
But with \verb|\mathclap|::
\[ \lim_{\mathclap{n \to +\infty}}x(n) \]
Using \verb|\scriptstyle| to resize:
\[ \lim_{\scriptscriptstyle n \to +\infty} x(n) \]
Using \verb|\Scale|
\[ \lim_{\Scale[0.5]{n \to +\infty}} x(n) \]
Using \verb|\Resize|
\[ \lim_{\Resize{\widthof{$\lim{}$}}{n \to +\infty}} x(n) \]
\end{document}

Answer 1

This doesn't work in subscripts or superscripts (it might be made so):

\documentclass{article}

\makeatletter
\newcommand{\awfullim}{\@ifnextchar_{\@awfullim}{\lim}}
\newcommand{\@awfullim}[2]{% #1 is _
  \settowidth{\dimen0}{$\lim$}%
  \settowidth{\dimen2}{$\scriptstyle#2$}%
  \ifdim\dimen2<\dimen0
    \lim_{#2}%
  \else
    \addtolength{\dimen2}{-\dimen0}%
    \kern-\dimen2 \lim_{\kern\dimen2 #2}%
  \fi
}
\makeatother

\begin{document}
\[
\awfullim_{x\to0^+}f(x)=\awfullim_{y\to+\infty}f(1/y)
\]
\end{document}

The name I used tells you how much I like this idea. ;-)

Answer 2

Stacks make this very easy. The alignment is set with a mode parameter (\stackalignment), the gap is set with an optional stacking argument. The use of \useanchorwidth says to not allow the width of the underset to affect the spacing to the next item. EDITED to use \mathop.

\documentclass{article}
\usepackage[usestackEOL]{stackengine}
\usepackage{graphicx}
\stackMath
\def\stackalignment{l}\def\useanchorwidth{T}
\begin{document}
\[
\mathop{\stackunder[3pt]{\lim}{\scriptstyle n\rightarrow+\infty}} x(n)
\]
\[
\mathop{\stackunder[2.5pt]{\lim}{\scriptscriptstyle n\rightarrow+\infty}} x(n)
\]
\[
\mathop{\stackunder[2pt]{\lim}{\scalebox{.33}{$n\rightarrow+\infty$}}} x(n)
\]
\end{document}

If \usearchorwidth had been defined as {F} instead of {T}, the result would look like

---

The alignment and anchorwidth parameters can be directly incorporated into the syntax, which has the further advantage of making their invocation impervious to the current mode settings.

\documentclass{article}
\usepackage[usestackEOL]{stackengine}
\usepackage{graphicx}
\stackMath
\begin{document}
\[
\mathop{\stackengine{3pt}{\lim}{\scriptstyle n\rightarrow+\infty}{U}{l}{F}{T}{S}} x(n)
\]
\[
\mathop{\stackengine{2.5pt}{\lim}{\scriptscriptstyle n\rightarrow+\infty}{U}{l}{F}{T}{S}} x(n)
\]
\[
\mathop{\stackengine{2pt}{\lim}{\scalebox{.33}{$n\rightarrow+\infty$}}{U}{l}{F}{T}{S}} x(n)
\]
\end{document}