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Theorem 12.2 Limit
Laws
for Functions
of
Two Variables
Let Land M
be
real numbers and suppose
that
limcx,y)~(a,b)
f(x,
y)
=Land
limcx,y)~(a,b)
g(x,y)
= M. Assume
cis
a constant and m and n are integers
1.
Sum
limcx,y)~(a,b)[f(x,y)
+
g(x,y)]
= L + M
2.
Difference
limcx,y)~(a,b)[f(x,y)
g(x,y)]
=
L
M
3.
Constant multiple
limcx,y)~(a,b)[cf(x,y)]
=
cL
4.
Product
limcx,y)~(a,b)
f(x,y)g(x,y)
=
LM
5 Q · l'
[f(x,y)]
L 'd d M 0
.
uot1e~t
Im(x,y)~(a,b)
g(x,y)
= M
provl
e
=f::.
6.
Power
limcx,y)~(a,b)[f(x,y)]n
=
Ln
7.
m/n
power
If
m and n have no common factors and
n;tQ
then
m m
limcx,y)~(a,b)
[f(x,
y)]n
= Ln where we assume
L>O
if
n
is
even
)
1
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