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How to prove that the limit of f(x,y,z) = xyz^2/(x^4 + y^2 + z^4) as (x,y,z) goes to (0,0,0) does not exist.
marcdelbes
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Jun 21, 2022
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How to prove that the limit of f(x,y,z) = xyz^2/(x^4 + y^2 + z^4) as (x,y,z) goes to (0,0,0) does not exist.
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marcdelbes
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Jun 21, 2022
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Since we can do AM-GM: x^(4)+y^(2)+z^(4)=x^(4)+y^(2)/2+y^(2)/2+z^(4)/2+z^(4)/2>=5(x^(4)y^(4)z^(8)/16)^(1/5)
It seems to me the limit is zero. ~~Maybe there is a typo that it's x^(2) not x^(4)?~~ nope x^(2) won't save this either. Not able to guess what the typo would be.
37games_beauty
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Jun 21, 2022
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I can't suggest a general method, but in this case note that the quantity yz\^2/(y\^2 + z\^4) is bounded in absolute value by 1/2, hence |f(x,y,z)| is bounded above by |x|/2.
plus_audition
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Jun 21, 2022
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Default would be spherical coordinates. You'll find the limit does exist.
inquiringshow
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Jun 21, 2022
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