Is it possible to redefine the limit operator to print Lim, not lim?

Question

The default limit operator displays “lim” with a small l. I wish to redefine it to display “Lim” that is, with a capital L like shown below.

Answer 1

You can add \ShowCommand\lim to your preamble to see the definition of \lim. It'll show

> \lim=robust macro:
->\protect \lim  .

> \lim =\long macro:
->\mathop {\operator@font lim}.
<argument> \lim  
                 
l.3 \ShowCommand\lim

You can then define something similar for \Lim or redefine \lim accordingly:

\makeatletter
\NewDocumentCommand{\Lim}{}{\mathop{\operator@font Lim}}% Similar to \lim
\RenewDocumentCommand{\lim}{}{\mathop{\operator@font Lim}}% Replace \lim (lim -> Lim)
\makeatother

Note that this default definition differs from how its defined by other packages. For example, with amsmath, the definition is

> \lim=\protected macro:
->\qopname \relax m{lim}.
<argument> \lim 
                
l.4 \ShowCommand\lim

where \qopname takes 3 arguments, the first of which is the operator text. You could

\NewDocumentCommand{\Lim}{}{\qopname{\relax}{m}{Lim}}
\RenewDocumentCommand{\lim}{}{\qopname{\relax}{m}{Lim}}

Best would be to use other tools (like mathtools), which help you define operators in a natural way:

\usepackage{mathtools}

\let\lim\relax% Remove existing \lim definition
\DeclareMathOperator*{\lim}{Lim}% (Re)define \lim
\DeclareMathOperator*{\Lim}{Lim}% Define \Lim

Here's a complete minimal example:

\documentclass{article}

%\ShowCommand\lim

%\makeatletter
%\NewDocumentCommand{\Lim}{}{\mathop{\operator@font Lim}}
%\RenewDocumentCommand{\lim}{}{\mathop{\operator@font Lim}}
%\makeatother


\usepackage{mathtools}

\let\lim\relax
\DeclareMathOperator*{\lim}{Lim}
\DeclareMathOperator*{\Lim}{Lim}

\begin{document}

\[
  \lim_{n \rightarrow \infty} S_n
  \qquad
  \Lim_{n \rightarrow \infty} S_n
\]

\end{document}

Answer 2

You can either define a new \Lim or redefine the existing \lim. Both are shown in this MWE.

\documentclass{article}
\usepackage{mathtools}
\DeclareMathOperator*{\Lim}{Lim}
\let\lim\relax
\DeclareMathOperator*{\lim}{Lim}
\begin{document}
$ \lim_{n\rightarrow\infty} S_n = \Lim_{n\rightarrow 0} Z_n$

\[ \lim_{n\rightarrow\infty} S_n = \Lim_{n\rightarrow 0} Z_n\]
\end{document}

Answer 3

The simplest way is to use amsmath (of course you already have it in a math document) and do

\renewcommand{\lim}{\operatorname*{Lim}}

This is (slightly) less efficient than doing

\let\lim\relax %%% undefine \lim
\DeclareMathOperator*{\lim}{Lim}

See also How to redefine a command using \DeclareMathOperator

Another strategy is, of course, to do

\DeclareMathOperator*{\Lim}{Lim}

and to use \Lim.