Iterating Limit

vector-calculus

Definition

Iterating Limit

Iterating limits are used in multivariate calculus when evaluating the limit of a function of two or more variables. Instead of approaching the limit point along an arbitrary path in the domain.

Example: Given a function $f:{\mathbb{R}}^2\to \mathbb{R}$ that takes in two variables.

Possible iterating limits of this function are:

• $L_1={\lim }_{x\to x_0}{\lim }_{y\to y_0}f(x,y)$
• $L_2={\lim }_{y\to y_0}{\lim }_{x\to x_0}f(x,y)$

If the overall limit $L={\lim }_{(x,y)\to (x_0,y_0)}f(x,y)$ and the inner one-dimensional limits exist, then:

$$L=L_1=L_2$$

Note that it’s still possible that if $L_1=L_2$ that $L$ may not exist.