Answer
limit does not exist
Work Step by Step
Given: $\lim\limits_{(x,y) \to (0,0)}f(x,y)=\frac{xy^{4}}{ {x^{2}+y^{8}}}$
We notice that if we directly substitute limits in the given function $f(x,y)=\frac{xy^{4}}{ {x^{2}+y^{8}}}$
Then $f(0,0)=\frac{0}{0}$
To evaluate limit along x-axis; put $y=0$
$f(x,0)=\frac{x.0^{4}}{ {x^{2}+0^{8}}}=0$
To evaluate limit along y-axis; put $x=0$
$f(0,y)=\frac{0.y^{4}}{ {0^{2}+y^{8}}}=0$
Approach (0,0) along another line let us say $x=y^{4}$
$f(y^{4},y)=\frac{y^{4}.y^{4}}{ {y^{8}+y^{8}}}$
Thus,
$\lim\limits_{(x,y) \to (0,0)}f(x,y)=\lim\limits_{(x,y) \to (0,0)}\frac{y^{4}.y^{4}}{ {y^{8}+y^{8}}}$
$\lim\limits_{y \to 0}\frac{y^{8}}{ 2y^{8}}$
$=\frac{1}{2}$
For a limit to exist, all the paths must converge to the same point. Since function $f(x,y)$ has two different values along two different paths, it follows that limit does not exist,
Hence, the limit does not exist.
