Math 251-copyright Joe Kahlig, 15A
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Section 12.2: Limits and Continuity
The notation
lim
(x,y)→(a,b)
f (x, y) = L says that as the point (x, y) gets arbitrarily close to (a, b), then
the difference of f (x, y) and L will be sufficiently small (i.e. very close to zero). Note: f (a, b) may not
be be defined.
If f (x, y) → L1 as (x, y) → (a, b) along path C1 and f (x, y) → L2 as (x, y) → (a, b) along path C2 ,
where L1 6= L2 then
lim f (x, y) does not exist.
(x,y)→(a,b)
Definition: A function f (x, y) is continuous at the point (x0 , y0 ) if
lim
(x,y)→(x0 ,y0 )
f (x, y) = f (x0 , y0 )
Definition: A function is said to be continuous on its domain if it is continuous at every (a, b) in
its domain.
Using the properties of limits, we find that sums, differences, products, and differences of continuous
functions are also continuous on their domain.
Example: Compute these limits.
(a)
lim
(x,y)→(3,2)
3x + 4xy + y 2 =
(b)
x4 −y 4
2
2
(x,y)→(2,2) x −y
(c)
x2 −3y 2
2
2
(x,y)→(0,0) x +y
lim
lim
=
=
Math 251-copyright Joe Kahlig, 15A
(d)
xy 2
2 +y 4
x
(x,y)→(0,0)
=
(e)
3x2 y
2
2
x
(x,y)→(0,0) +y
=
lim
lim
Example: Determine The largest set on which the function is continuous.
f (x, y) = arcsin(x2 + y 2 )
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