Iterated limit

In multivariable calculus, an iterated limit is a limit of a sequence or a limit of a function in the form

or other similar forms.

An iterated limit is only defined for an expression whose value depends on at least two variables. To evaluate such a limit, one takes the limiting process as one of the two variables approaches some number, getting an expression whose value depends only on the other variable, and then one takes the limit as the other variable approaches some number.

This section introduces definitions of iterated limits in two variables. These may generalize easily to multiple variables.

For each ${\displaystyle n,m\in \mathbf {N} }$, let ${\displaystyle a_{n,m}\in \mathbf {R} }$ be a real double sequence. Then there are two forms of iterated limits, namely

For example, let

Then

Let ${\displaystyle f:X\times Y\to \mathbf {R} }$. Then there are also two forms of iterated limits, namely