Answer
limit does not exist
Work Step by Step
Given: $\lim\limits_{(x,y,z) \to (0,0,0)}\frac{xy+yz^{2}+xz^{2}}{ {x^{2}+y^{2}+z^{4}}}$
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 $y=0,z=0$
$=\lim\limits_{(x,y,z) \to (0,0,0)}\frac{0+0+0}{ {x^{2}+0^{2}+0^{4}}}=\frac{0}{x^{2}}=0$
Approach the point (0, 0, 0) along the curve where $x=y$ and $x=z^{2}$
$\lim\limits_{(x,y,z) \to (0,0,0)}\frac{x^{2}+x^{2}+x^{2}}{ {x^{2}+x^{2}+x^{2}}}=\frac{0}{x^{2}}$
$=\lim\limits_{(x,y,z) \to (0,0,0)}\frac{3x^{2}}{3x^{2}}$
$=1$
For a limit to exist, all the paths must converge to the same point.Since, function $f(x,y,z)$ has two different values along two different paths, it follows that limit does not exist,
Hence, the limit does not exist.
