Question

Find the limit, if it exists. (if an answer does not exist, enter dne.) lim (x, y, z)→(0, 0, 0) xy + 3yz2 + 4xz2 x2 + y2 + z4

Answer 1

Consider what happens along the line through the origin with $x=y=z$. The limit reduces to

$\displaystyle\lim_{(x,y,z)\to(0,0,0)}\frac{xy+3yz^2+4xz^2}{x^2+y^2+z^4}=\lim_{x\to0}\frac{x^2+7x^3}{2x^2+x^4}=\lim_{x\to0}\frac{1+7x}{2+x^2}$

but this limit does not exist.