Limits of function of two variables
Limits of Functions of Two Variables Limits of Functions of Two Variables Definition Let f(x,y) be defined in a neighborhood of the point (a,b) , except possibly at the point itself. We say the limit of f(x,y) as (x,y) → (a,b) is L, and write: lim (x,y)→(a,b) f(x,y) = L if for every ε>0, there exists δ>0 such that: 0 < √((x-a)² + (y-b)²) < δ ⇒ |f(x,y)-L| < ε Intuitive Explanation The value of f(x,y) gets closer to L as (x,y) approaches (a,b) from any direction. The limit must be the same along all paths approaching (a,b). Examples Example 1 Function: f(x,y) = x + y Find: lim (x,y)→(1,2) f(x,y) Solution: f(1,2) = 1 + 2 = 3 Answer: 3 Example 2 Function: f(x,y) = xy / (x² + y²) Find: lim (x,y)→(0,0) f(x,y) Solution: Along y=0: f(x,0)=0 Along x=0: f(0,y)=0 Along y=x: f(x,x)=1/2 Answer: Limit does not exist (different values along different paths) Example 3 Function: f(x,y) = x² + y² Find: lim (x,y)→(1,1) f(x,y) ...