15.2 Limits and Continuity

Assigned homework: 7,9,13,17,21,27,29,31,33,35,37, 41,47,51,54,57,61,63,67,71,73,77,78,83

Limit of a Function of Two Variables

• Condition $|PP_0|< \delta$ means that the distance bewtween $P(x,y)$ and $P_0(a,b)$ is less than $\delta$ as $P$ approaches $P_0$ from all possible directions

Limits at Boundary Points

If R is a region in $\R^2$, an interior point of P of R lies entirely within R, which means it is possible to find a disk centered at P that contains only points of R.

Continuity of Functions of Two Variables

A function is continuous at any point on the region for which it is defined

1. Polynomials are continuous everywhere
2. Rational functions are continuous everywhere the denominator is not 0
3. Continuity holds for compositions of continuous functions

Just because the partials exist, doesn’t mean the function is diffrentiable.

Where should I take the limit?

Refers to the two-path tets for nonexistence of limits

• Along $x = 0$, $y = 0$, $x = y$, etc
• Family of linear paths $y = mx$ or $x = my$