Section 10.2: 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 equal to L if f (x, y) can be made arbitrarily
close to L by taking (x, y) sufficiently close to (a, b). This is denoted as
lim
f (x, y) = L.
(x,y)→(a,b)
Each of the limit laws from Chapter 3 applies to limits of functions of multiple variables.
Example: Calculate each limit.
(a)
lim
(2xy + 3x2 )
(x,y)→(−1,1)
(b)
lim
2xy
+ y2
(x,y)→(1,1) x2
Note: Recall that lim f (x) exists if and only if
x→a
lim f (x) = lim+ f (x).
x→a−
x→a
For functions of one variable, we can only approach x = a from two directions. However, for
functions of two variables, there are infinitely many ways to approach (a, b) in the plane.
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Theorem: If f (x, y) approaches L1 along a path P1 and f (x, y) approaches L2 along a path
P2 as (x, y) → (a, b), where L1 6= L2 , then lim f (x, y) does not exist.
(x,y)→(a,b)
Example: Find the limit
x2 − y 2
(x,y)→(0,0) x2 + y 2
lim
if it exists.
Example: Find the limit
lim
(x,y)→(0,0) x2
if it exists.
2
2xy
+ 2y 2
Example: Find the limit
x2 y
lim
(x,y)→(0,0) x4 + y 2
if it exists.
Example: Find the limit
x2 − xy + x − y
(x,y)→(0,0)
x−y
lim
if it exists.
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Definition: A function f (x, y) is continuous at (a, b) if
lim
f (x, y) = f (a, b).
(x,y)→(a,b)
In order for f to be continuous at (a, b), the following conditions must hold:
1. f (a, b) is defined
2.
lim
f (x, y) exists
(x,y)→(a,b)
3.
lim
f (x, y) = f (a, b).
(x,y)→(a,b)
Example: Show that
f (x, y) =



3xy
, (x, y) 6= (0, 0)
+ y2
0,
(x, y) = (0, 0)
x2
is discontinuous at (0, 0).
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