limit(x,y)->(0,0) (1-|x|-|y|)^(xy/(x^2+y^2))

[jazzman]

Hey guys!
I'm having trouble with this 2-variable limit:

\(\displaystyle \lim_{(x,y)\to(0,0)}{(1-|x|-|y|)^\frac{xy}{x^2+y^2}}\)

I know that
\(\displaystyle \left|{\frac{xy}{x^2+y^2}}\right|\le{\frac{1}{2}}\)
and that
\(\displaystyle \lim_{(x,y)\to(0,0)}{\frac{xy}{x^2+y^2}}\) doesn't exist.

Please help.

 

[dts5044]

you need to use the old e^(lnu) trick to get rid of your exponent:

lim (1 - |x| - |y|)^(xy/(x^2 + y^2))
(x,y) --> (0,0)

=
lim xy/(x^2 + y^2) * ln(1 - |x| - |y|)
(x,y) --> (0,0)
e^

now, we know that: ln(1 - |x| - |y|) <= 0

provided |x| + |y| < 1

try and use this to come up with appropriate bounds for applying the squeeze theorem