Variant limits
Variants for superior, inferior, projective, and injective limits are an interesting challenge for typesetting engines with mathematical capabilities.
They have both a subscript expression and a stretchy symbol (an arrow or a bar) that indicates the direction and nature of the limit.
The symbol is stacked below or above the limit.
In display mode, the expression is below the limit.
In inline mode, the expression is on the right of the limit, but the symbol remains above or below the limit. So the symbol is not moveable, while the expression is, depending on the context.
\begin[papersize=a7]{document}
\nofolios
\neverindent
\use[module=packages.math]
\font[size=12pt]
\set[parameter=math.font.family,value=Libertinus Math]
\set[parameter=math.font.size,value=12]
\begin[mode=display]{math}
\projlim_{i \in I} A_i = \varprojlim_{i \in I} A_i
\end{math}
Inline: \math{\projlim_{i \in I} A_i = \varprojlim_{i \in I} A_i}
\begin[mode=display]{math}
\injlim_{i \in I} A_i = \varinjlim_{i \in I} A_i
\end{math}
Inline: \math{\injlim_{i \in I} A_i = \varinjlim_{i \in I} A_i}
\begin[mode=display]{math}
\limsup_{n \to \infty} a_n = \varlimsup_{n \to \infty} a_n
\end{math}
Inline: \math{\limsup_{n \to \infty} a_n = \varlimsup_{n \to \infty} a_n}
\begin[mode=display]{math}
\liminf_{n \to \infty} a_n = \varliminf_{n \to \infty} a_n
\end{math}
Inline: \math{\liminf_{n \to \infty} a_n = \varliminf_{n \to \infty} a_n}
\end{document}
$ sile -o varlimits-sile.pdf varlimits-sile.sil
\documentclass[12pt]{article}
\usepackage[paperheight=105mm,paperwidth=74mm,margin=4mm]{geometry}
\pagenumbering{gobble}
\setlength{\parindent}{0pt}
\usepackage{libertinus}
\usepackage{fontspec}
\usepackage{unicode-math}
\begin{document}
$$
\projlim_{i \in I} A_i = \varprojlim_{i \in I} A_i
$$
Inline: $\projlim_{i \in I} A_i = \varprojlim_{i \in I} A_i$
$$
\injlim_{i \in I} A_i = \varinjlim_{i \in I} A_i
$$
Inline: $\injlim_{i \in I} A_i = \varinjlim_{i \in I} A_i$
$$
\limsup_{n \to \infty} a_n = \varlimsup_{n \to \infty} a_n
$$
Inline: $\limsup_{n \to \infty} a_n = \varlimsup_{n \to \infty} a_n$
$$
\liminf_{n \to \infty} a_n = \varliminf_{n \to \infty} a_n
$$
Inline: $\liminf_{n \to \infty} a_n = \varliminf_{n \to \infty} a_n$
\end{document}
$ lualatex --interaction=batchmode --halt-on-error --jobname=data/varlimits-lualatex varlimits-lualatex.tex
// 2025, Didier Willis
// License: MIT.
// Typst does not have these extended limits out of the box:
// projlim, injlim, varprojlim, varinjlim,
// varlimsup, varliminf.
// So here is a possible implementation...
// Total newbie, so it's maybe not idiomatic
#let varlim(arrow) = context {
let lim = "lim"
let lim-width = measure(lim).width
let arrow-height = measure(arrow).height
let varlimop = box(
width: lim-width,
baseline: arrow-height,
box(
width: 0pt,
baseline: arrow-height,
[#math.stretch(arrow, size: lim-width)],
)
+ box(
baseline: 0pt,
width: 0pt,
lim,
),
)
return varlimop
}
// Helpers
#let strut = context {
// Weird way to do a small adjustment for varliminf
// so that the limit expression is a bit lower
block(height: measure("").height, width: 0pt)
}
#let varprojlim = math.limits(varlim(sym.arrow.l), inline: false)
#let projlim = math.limits("proj\u{2009}lim", inline: false)
#let varinjlim = math.limits(varlim(sym.arrow.r), inline: false)
#let injlim = math.limits("inj\u{2009}lim", inline: false)
#let varliminf = math.limits(underline("lim" + strut), inline: false)
#let varlimsup = math.limits(overline("lim"), inline: false)
// --- Now the intended document ---
#set page(
paper: "a7",
margin: 4mm,
)
#set text(
font: "Libertinus Serif",
size: 12pt,
)
$ projlim_(i in I) A_i = varprojlim_(i in I) A_i $
Inline: $projlim_(i in I) A_i = varprojlim_(i in I) A_i$
$ injlim_(i in I) A_i = varinjlim_(i in I) A_i $
Inline: $injlim_(i in I) A_i = varinjlim_(i in I) A_i$
$ limsup_(n -> oo) a_n = varlimsup_(n -> oo) a_n $
Inline: $limsup_(n -> oo) a_n = varlimsup_(n -> oo) a_n$
$ liminf_(n -> oo) a_n = varliminf_(n -> oo) a_n $
Inline: $liminf_(n -> oo) a_n = varliminf_(n -> oo) a_n$
$ typst compile varlimits-typst.typ varlimits-typst.pdf
<html>
<head>
<style>
@page { size: A7; margin: 4mm; }
body { font-family: "Libertinus Serif"; font-size: 11pt; }
math { font-family: "Libertinus Math"; }
p { margin: 0.4em 0; }
</style>
</head>
<body>
<p><math display="block"><mrow>
<munder><mrow><mi>proj lim</mi></mrow><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder>
<msub><mi>A</mi><mi>i</mi></msub>
<mo>=</mo>
<munder>
<munder><mi>lim</mi><mo stretchy="true">←</mo></munder>
<mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow>
</munder>
<msub><mi>A</mi><mi>i</mi></msub>
</mrow></math></p>
<p>Inline: <math><mrow>
<msub><mrow><mi>proj lim</mi></mrow><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub>
<msub><mi>A</mi><mi>i</mi></msub>
<mo>=</mo>
<msub>
<munder><mi>lim</mi><mo stretchy="true">←</mo></munder>
<mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow>
</msub>
<msub><mi>A</mi><mi>i</mi></msub>
</mrow></math>, the arrow stays under the limit while the
expression moves to the side.</p>
<p><math display="block"><mrow>
<munder><mrow><mi>inj lim</mi></mrow><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder>
<msub><mi>A</mi><mi>i</mi></msub>
<mo>=</mo>
<munder>
<munder><mi>lim</mi><mo stretchy="true">→</mo></munder>
<mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow>
</munder>
<msub><mi>A</mi><mi>i</mi></msub>
</mrow></math></p>
<p><math display="block"><mrow>
<munder><mrow><mi>lim inf</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></munder>
<msub><mi>a</mi><mi>n</mi></msub>
<mo>=</mo>
<munder>
<munder><mi>lim</mi><mo stretchy="true">̲</mo></munder>
<mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow>
</munder>
<msub><mi>a</mi><mi>n</mi></msub>
</mrow></math></p>
<p>varlimsup (lim with an overline) is omitted: Glu stretches the
overline to the right width but does not yet place it tight above
its base.</p>
</body>
</html>
$ glu --css all-system-fonts.css -o varlimits-glu.pdf varlimits-glu.html