The commonest hand-written style for expressions is to place the limit
expressions on operators such as `\`

`sum`

and `\`

`int`

physically
above and below the operator. In (La)TeX, we write these limit
expressions using sub- and superscripts applied to the operator, but
they don’t always appear in the “handwritten” way in TeX’s
output.

The reason is, that when an expression appears in non-display maths,
in running text (and is therefore in TeX `\`

`textstyle`

), placing
the limits thus could lead to ragged line spacing (and hence
difficult-to-read text). It is therefore common (in `\`

`textstyle`

)
to place the limits as one would sub- and superscripts of variables.

This is not universally satisfactory, so the primitive `\`

`limits`

is
provided:

$\sum\limits_{n=1}^{m} ...$

which will place the limits right above and below the symbol (and be blowed to the typography…).

Contrariwise, you may wish to change the arrangement of the limits
when in `\`

`displaystyle`

. For this purpose, there’s a corresponding
`\`

`nolimits`

:

\[\sum\nolimits_{n=1}^{m} ...\]

which will place the limits as they would be in `\`

`textstyle`

.

Alternatively, one can manipulate the
`\`

`textstyle`

/`\`

`displaystyle`

state of the mathematics. To get
“`\`

`limits`

placement” in inline maths,

$\displaystyle\sum_{n=1}^{m} ...$

and for “`\`

`nolimits`

placement” in display maths,
`\`

`nolimits`

:

\[\textstyle\sum_{n=1}^{m} ...\]

will serve. Either of these forms may have effects other than on the operator you’re considering, but there are still those who prefer this formulation.

Remember, if you’re declaring a special operator of your own, the AMSLaTeX functions (that you ought to be using) allow you to choose how limits are displayed, at definition time.

(Note that the macro `\`

`int`

normally has `\`

`nolimits`

built in to
its definition. There is an example in the TeXbook to show how odd
`\`

`int`

`\`

`limits`

looks when typeset.)

This question on the Web: http://www.tex.ac.uk/cgi-bin/texfaq2html?label=limits