<s>[INST] Limits in multivariable Calculus $$\lim_{(x,y,z)\to(0,0,0)} \dfrac{(x+y+z)^2}{x^2 + y^2 + z^2}$$
According to me the answer should be $3$ but it says limit does not exist. Pls let me know where am I wrong.
The expression simplifies to $$1 + \dfrac{2\cdot(xy+yz+zx)}{x^2+y^2+z^2}$$
And for $x=y=z$ (say  $x = y = z = a$) we have 
$1+ 2\cdot \left(\dfrac{3a^2}{3a^2}\right) 
= 1+2 =3$. 
 [/INST] You have determined the limit for the special case $x=y=z=t \to 0$ but for $x+y+z=0$, that is for example $x=t$, $y=-t-t^2$, $z=t^2$ we obtain
$$\frac{(x+y+z)^2}{x^2+y^2+z^2}=0$$
thereofore the limit doesn't exist.
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