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:

- snippet.latex
`$\sum\limits_{n=1}^{m} ...$`

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`

:

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

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,

- snippet.latex
`$\displaystyle\sum_{n=1}^{m} ...$`

and for “`\nolimits`

placement” in display maths,
`\nolimits`

:

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

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.)