Finding the limit of $\frac{xy+yz+zx}{\sqrt{x^2+y^2+z^2}}$ as $(x,y,z)$ approach $(0,0,0)$
So I used polar coordinates and found the limit to be $0$. However I tried using successions $\{x_k\},\{y_k\},\{z_k\}$, who all approach 0, but I got stuck.
Can someone help, pls?
Thank you so much!
Solution 1:
$|\frac {xy} {\sqrt {x^{2}+y^{2}+z^{2}}}| \leq |\frac {xy} {\sqrt {x^{2}}}| =|y| \to 0$ Similarly, $|\frac {yz} {\sqrt {x^{2}+y^{2}+z^{2}}}| \to 0$ and $|\frac {zx} {\sqrt {x^{2}+y^{2}+z^{2}}}| \to 0$. Add these three.