Limits

iff:

The target point can be approached in any directions. Thus, if 2 different paths yield 2 different limtting values, then the limit doesn’t exist.

Multivariable limit properties are analogous to the single variable limits.

Uniqueness of limit

Let be a real-valued function defined on . Let . If limit of as approaches exists, then it is unique.

Proof Hint

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

Iterated limits

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

Similarily, another limit exists.

Both of the above limits are not necessarily equal. If exists, they will be equal.