Limits

iff:

In the above definition statement, circular disk was used; square neighbourhood can also be used. The target point can be approached in any directions.

Multivariable limit properties are analogous to the single variable limits.

Uniqueness of limit

Let be a real-valued function defined on . Let .

If exists, then it is unique.

Proof Hint

• Consider 2 limits for at
• Consider for both limitting values for a chosen
• Reduce to

Non existence of limit

Suppose as , along a path and along a different path .

If then the doesn’t exist.

Iterated limits

Aka. repeated limits. If is defined in a neighborhood of a point in and exists, which is a function of only, then the limit of this function as can be written as:

Similarily, another limit exists.

Note:

• The two repeated limits may or may not exist
• The two repeated limits, when they exist, may or may not be equal.
• The existence of the 2-variable limit does not imply the existence of either of the two repeated limits
• The existence of the repeated limits does not imply the existence of 2-variable limit
• If the repeated limit exists and they are not equal, then the 2-variable limit cannot exist.
• If a repeated limit exists, along with the 2-variable limit, these two would be equal