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.

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.
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.
Default would be spherical coordinates. You'll find the limit does exist.

0 like 0 dislike