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.
This section introduces various definitions of limits in two variables.
These may generalize easily to multiple variables.
[6] For this limit to exist, f(x, y) can be made as close to L as desired along every possible path approaching the point (a, b).
In this definition, the point (a, b) is excluded from the paths.
The other type is the double limit, denoted by which means that for all
[7] For this limit to exist, f(x, y) can be made as close to L as desired along every possible path approaching the point (a, b), except the lines x=a and y=b.
In other words, the value of f along the lines x=a and y=b does not affect the limit.
Combining Theorem 2 and 3, we have the following corollary: For a two-variable function
, we may also define the double limit at infinity which means that for all
Similar definitions may be given for limits at negative infinity.
The following theorem states the relationship between double limit at infinity and iterated limits at infinity: For example, let Since
The converses of Theorems 1, 3 and 4 do not hold, i.e., the existence of iterated limits, even if they are equal, does not imply the existence of the double limit.
A sufficient condition for interchanging limits is given by the Moore-Osgood theorem.
[8] The essence of the interchangeability depends on uniform convergence.
A corollary is about the interchangeability of infinite sum.
Similar results hold for multivariable functions.
Note that this theorem does not imply the existence of
[10] An important variation of Moore-Osgood theorem is specifically for sequences of functions.
A corollary is the continuity theorem for uniform convergence as follows: Another corollary is about the interchangeability of limit and infinite sum.
The explanation for this paradox is that the vertical sum to infinity and horizontal sum to infinity are two limiting processes that cannot be interchanged.
By the integration theorem for uniform convergence, once we have
, the limit in n and an integration over a bounded interval
can be interchanged: However, such a property may fail for an improper integral over an unbounded interval
In this case, one may rely on the Moore-Osgood theorem.
Then by the integration theorem for uniform convergence,
, the Moore-Osgood theorem requires the infinite series to be uniformly convergent.