In multivariable calculus, an iterated limit is an expression of the form
Contents
One has an expression whose value depends on at least two variables, one takes the limit 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 is not defined in the same way as the limit
which is not an iterated limit. To say that this latter limit of a function of more than one variable is equal to a particular number L means that ƒ(x, y) can be made as close to L as desired by making the point (x, y) close enough to the point (p, q). It does not involve first taking one limit and then another.
Counterexamples
It is not in all cases true that
Among the standard counterexamples are those in which
and
and (p, q) = (0, 0).
In the first example, the values of the two iterated limits differ from each other:
and
In the second example, the two iterated limits are equal to each other despite the fact that the limit as (x, y) → (0, 0) does not exist:
and
but the limit as (x, y) → (0, 0) along the line y = x is different:
It follows that
does not exist.
Sufficient condition
A sufficient condition for(1) to hold is Moore-Osgood theorem: If