# How do you put multiple things under a limit?

I'm reading about inverse limits, and the notation sometimes requires some sort of index or set over which to take the inverse limit. It looks something like this:

How would I correctly TeX that? I tried something like {\lim_{\longleftarrow}}_{H\in\mathscr{F}} but the arrow and the set are side by side, not one over the other.

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There is no need for \substack in this case: there are the \varinjlim and \varprojlim macros taking care of both direct and inverse limits, putting a subscript now results in the behaviour you're looking for (and everyone is mimicking using \substack).

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Thanks Pieter, I didn't know such macros existed. Just was I was looking for. –  yunone Aug 19 '11 at 19:19
@Pieter: You're right. Use \usepackage{mathtools} and then \underset{i}{\varinjlim} for inverse limit, and \underset{H\in\mathscr{F}}{\varprojlim} for direct limit (the latter requires \usepackage{mathrsfs} as well). –  Werner Aug 19 '11 at 19:28
@Werner He needs \varprojlim when reproducing the examples, inverse limits are also known as projective limits. And why use \underset? Just \varprojlim_i will do fine, even in text style where it is appropriate for the limit to be on the right. –  Pieter Aug 19 '11 at 19:33
@Pieter: You're right, again... –  Werner Aug 19 '11 at 19:39

Without going into too much font detail, you can use the \mathop{<stuff>} command to make an operator of almost anything in mathmode. For example, here's a small modification to your initial try:

\mathop{\lim_{\longleftarrow}}_{i}G/H_i \approx
\mathop{\lim_{\longleftarrow}}_{H\in\mathcal{F}}G/H


However, you may also be interested in stacking items on top of one another using \stackrel{<top>}{<bottom>}:

\lim_{\stackrel{\longleftarrow}{i}}G/H_i \approx
\lim_{\stackrel{\longleftarrow}{H\in\mathcal{F}}}G/H


Consider browsing through Herbert Voß' mathmode document. It is filled with typesetting information in mathmode.

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Thanks Werner, \stackrel seems to work quite nicely. –  yunone Aug 19 '11 at 18:54
@yunone: It does modify the fontsize slightly for the <upper> part. In this case \longleftarrow; that's why you see it being slightly shorter. –  Werner Aug 19 '11 at 18:56

You could use \substack of amsmath:

\lim_{\substack{\longleftarrow\\H\in\mathscr{F}}}


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\documentclass{article}
\usepackage{mathtools}
\begin{document}

$\lim_{\xleftarrow[i]{}}G/H_i \approx \lim_{\xleftarrow[\mathclap{H\in\mathcal{F}}]{}}G/H$

\end{document}


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You can define a new operator using \DeclareMathOperator of the amsmath package as follows:

\documentclass{article}

\usepackage{mathtools}

\DeclareMathOperator*{\ulalim}{\underleftarrow\lim}

\begin{document}

$\ulalim_{i}G/H_{i}\approx\smashoperator{\ulalim_{H\in\mathcal{F}}}G/H$

\end{document}


The resulting output is:

Note the usage of \smashoperator to remove redundant white space between the operator and the operand on the right-hand side of the equation.

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OK, in this specific case, you actually don't need to define a new operator since there is already one as pointed out by @Pieter. –  mhp Aug 19 '11 at 20:09