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VTRMC
2005 VTRMC
Problem 5
limit in two variables
limit in two variables
Source: VTRMC 2005 P5
June 10, 2021
limits
real analysis
Problem Statement
Define
f
(
x
,
y
)
=
x
y
x
2
+
y
2
ln
(
x
2
)
2
f(x,y)=\frac{xy}{x^2+y^2\ln(x^2)^2}
f
(
x
,
y
)
=
x
2
+
y
2
l
n
(
x
2
)
2
x
y
if
x
≠
0
x\ne0
x
=
0
, and
f
(
0
,
y
)
=
0
f(0,y)=0
f
(
0
,
y
)
=
0
if
y
≠
0
y\ne0
y
=
0
. Determine whether
lim
(
x
,
y
)
→
(
0
,
0
)
f
(
x
,
y
)
\lim_{(x,y)\to(0,0)}f(x,y)
lim
(
x
,
y
)
→
(
0
,
0
)
f
(
x
,
y
)
exists, and find its value is if the limit does exist.
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