Section 12.2: Limits and Continuity
Definition: Let f be a function of two variables defined on a disk containing (a, b). Then the
limit of f (x, y) as (x, y) approaches (a, b) is L, written as
lim
f (x, y) = L,
(x,y)→(a,b)
if f (x, y) can be made arbitrarily close to L by taking (x, y) sufficiently close to (a, b).
Note: Recall that lim f (x) exists if and only if lim− f (x) = lim+ f (x). We can only approach
x→a
x→a
x→a
x = a from two directions. However, for functions of two variables, there are infinitely many
ways to approach (a, b) in the plane.
Figure 1: There are infinitely many paths along which (x, y) approaches (a, b).
Theorem: If f (x, y) → L1 along a path P1 and f (x, y) → L2 along a path P2 as (x, y) → (a, b)
where L1 6= L2 , then
lim f (x, y)
(x,y)→(a,b)
does not exist.
Example: Find the limit
x2 − y 2
lim
(x,y)→(0,0) x2 + y 2
if it exists.
Approaching (0, 0) along the x-axis (y = 0),
x2 − y 2
x2
=
lim
= 1.
x→0 x2
(x,y)→(0,0) x2 + 2y 2
lim
Approaching (0, 0) along the y-axis (x = 0),
x2 − y 2
−y 2
=
lim
= −1.
y→0 y 2
(x,y)→(0,0) x2 + 2y 2
lim
The limit does not exist.
Example: Find the limit
2xy
(x,y)→(0,0) x2 + 2y 2
lim
if it exists.
Approaching (0, 0) along the x-axis (y = 0),
lim
(x,y)→(0,0) x2
0
2xy
= lim 2 = 0.
2
x→0 x
+ 2y
Approaching (0, 0) along the y-axis (x = 0),
0
2xy
=
lim
= 0.
y→0 2y 2
(x,y)→(0,0) x2 + 2y 2
lim
Approaching (0, 0) along the line y = x,
2x2
2
2xy
=
lim
= .
2
2
2
x→0 3x
(x,y)→(0,0) x + 2y
3
lim
The limit does not exist.
Example: Find the limit
x2 y
(x,y)→(0,0) x4 + y 2
lim
if it exists.
Approaching (0, 0) along the line y = mx,
x2 y
mx3
lim
= lim 4
= 0.
x→0 x + m2 x2
(x,y)→(0,0) x4 + y 2
Approaching (0, 0) along the parabola y = x2 ,
x2 y
x4
1
=
lim
= .
4
2
4
x→0 2x
(x,y)→(0,0) x + y
2
lim
The limit does not exist.
Example: Find the limit
x2 − xy + x − y
(x,y)→(0,0)
x−y
lim
if it exists.
Factoring the numerator,
x2 − xy + x − y
=
(x,y)→(0,0)
x−y
x(x − y) + (x − y)
(x,y)→(0,0)
x−y
(x − y)(x + 1)
=
lim
(x,y)→(0,0)
x−y
=
lim (x + 1) = 1.
lim
lim
(x,y)→(0,0)
Definition: Let f be a function of two variables defined on a disk centered at (a, b). Then f
is continuous at (a, b) if
lim f (x, y) = f (a, b).
(x,y)→(a,b)
Note: Polynomials and rational functions are continuous at every point in their domain.
Example: Determine the set where the function

 3x2 y
, (x, y) 6= (0, 0)
f (x, y) =
x2 + y 2

3,
(x, y) = (0, 0)
is continuous.
The only suspicious point is (0, 0). Assuming the limit does exist, approaching (0, 0) along
the x-axis gives
3x2 y
0
lim
= lim 2 = 0 6= 3.
2
2
x→0 x
(x,y)→(0,0) x + y
The function is continuous on the set {(x, y) ∈ R2 |(x, y) 6= (0, 0)}.