Limits of function 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)

Solution: Direct substitution: f(1,1) = 1² + 1² = 2

Answer: 2

Example 4

Function: f(x,y) = (x² - y²) / (x² + y²)

Find: lim_{(x,y)→(0,0)} f(x,y)

Solution:

• Along y=0: f(x,0)=1
• Along x=0: f(0,y)=-1

Answer: Limit does not exist

Example 5

Function: f(x,y) = 3x²y / (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=kx: f(x,kx) → 0 as x→0

Answer: 0

Summary Table

Function	Point	Limit	Exists?
x + y	(1,2)	3	✔
xy / (x² + y²)	(0,0)	–	✘
x² + y²	(1,1)	2	✔
(x² - y²) / (x² + y²)	(0,0)	–	✘
3x²y / (x² + y²)	(0,0)	0	✔