Answer
$0$
Work Step by Step
Given: $\lim\limits_{(x,y,z) \to (0,0,0)}\frac{x^{2}y^{2}z^{2}}{ {x^{2}+y^{2}+z^{2}}}$
We can approach the points (0, 0, 0) in space through the co-ordinate axes or through co-ordinate plane or through the symmetrical or unsymmetrical lines.
Now, approach the point (0, 0, 0) along x-axis.
To evaluate limit along x-axis; put $y=0,z=0$
$f(x,0,0)=\frac{x^{2}0^{2}0^{2}}{ {x^{2}+0^{2}+0^{2}}}=0$
To evaluate limit along y-axis; put $x=0,z=0$
$f(0,y,0)=\frac{0^{2}y^{2}0^{2}}{ {0^{2}+y^{2}+0^{2}}}=0$
To evaluate limit along z-axis; put $x=0,y=0$
$f(0,0,z)=\frac{0^{2}0^{2}z^{2}}{ {0^{2}+0^{2}+z^{2}}}=0$
Hence, the limit converges to $0$.
