Limit of a function of 3 variables

Question

I need help to evaluate the following limit please, I used to use polar coordinates in most 2 variable functions but here I am stuck.

$\lim_{(x,y,z)\to(0,0,0)}$$\frac{xy+yz^2+xz^2}{x^2+y^2+z^2}$

well thankyou, found that the result is dependent on ($\theta\ and\ \phi$) and hence the limit does not exist. — mandez, Apr 15 '15 at 15:18

score 5 · Answer 1 · answered Apr 15 '15 at 15:18

Observe $$(x,y,z)\to (0,0,0)\quad\text{by the path}\quad(t,t,0)\;\quad\implies\;\quad\frac{xy+yz^2+xz^2}{x^2+y^2+z^2}\to\frac{1}{2}$$ $$(x,y,z)\to (0,0,0)\quad\text{by the path}\quad(0,0,t)\;\quad\implies\;\quad\frac{xy+yz^2+xz^2}{x^2+y^2+z^2}\to 0$$

Therefore, the limit doesn't exist.

Limit of a function of 3 variables

1 Answers1