When to use \nolimits when defining a math operator?

Question

In plain TeX, \cos is defined as \mathop {\rm cos}\nolimits but \gcd as \mathop {\rm gcd}.

In what case to use \nolimits when defining a math operator to be typeset in roman?

Answer 1

I'm afraid this is not a TeX question, but a math one. Anyway, this is the list in plain.tex:

\def\log{\mathop{\rm log}\nolimits}
\def\lg{\mathop{\rm lg}\nolimits}
\def\ln{\mathop{\rm ln}\nolimits}
\def\lim{\mathop{\rm lim}}
\def\limsup{\mathop{\rm lim\,sup}}
\def\liminf{\mathop{\rm lim\,inf}}
\def\sin{\mathop{\rm sin}\nolimits}
\def\arcsin{\mathop{\rm arcsin}\nolimits}
\def\sinh{\mathop{\rm sinh}\nolimits}
\def\cos{\mathop{\rm cos}\nolimits}
\def\arccos{\mathop{\rm arccos}\nolimits}
\def\cosh{\mathop{\rm cosh}\nolimits}
\def\tan{\mathop{\rm tan}\nolimits}
\def\arctan{\mathop{\rm arctan}\nolimits}
\def\tanh{\mathop{\rm tanh}\nolimits}
\def\cot{\mathop{\rm cot}\nolimits}
\def\coth{\mathop{\rm coth}\nolimits}
\def\sec{\mathop{\rm sec}\nolimits}
\def\csc{\mathop{\rm csc}\nolimits}
\def\max{\mathop{\rm max}}
\def\min{\mathop{\rm min}}
\def\sup{\mathop{\rm sup}}
\def\inf{\mathop{\rm inf}}
\def\arg{\mathop{\rm arg}\nolimits}
\def\ker{\mathop{\rm ker}\nolimits}
\def\dim{\mathop{\rm dim}\nolimits}
\def\hom{\mathop{\rm hom}\nolimits}
\def\det{\mathop{\rm det}}
\def\exp{\mathop{\rm exp}\nolimits}
\def\Pr{\mathop{\rm Pr}}
\def\gcd{\mathop{\rm gcd}}
\def\deg{\mathop{\rm deg}\nolimits}

and you can see that \nolimits is for symbols of function taking a fixed number of arguments (mostly one, but \hom takes two), whereas \limits is for functions with any number of arguments (or whose standard argument is a set).

Exception: the limit related symbols (which originated the others not taking \nolimits).

Answer 2

Trig functions are often used with superscripts

You can see why \nolimits are needed as you don't want limits

$$ \mathop{\rm cos}^2  + \mathop{\rm cos}^2  = 1$$

$$ \mathop{\rm cos}\nolimits^2  + \mathop{\rm cos}\nolimits^2  = 1$$

\bye

Answer 3

Use \nolimits when you do not want superscripts and subscripts to appear above and below the operator in display mode, the way they do for \lim, \sum or \int. An example is \cos^2 t, which should display as cos² t, and not with a 2 above the letter o.