Given $$ f(x,y) = \begin{cases} \frac{xy^3}{x+y^2}, & (x,y) \neq (0,0), \\ 0, & (x,y) = (0,0). \end{cases} $$
Now I computed $$f_{yx} = \frac{5xy^4+y^6}{(x+y^2)^3}$$
Now I approach from $y=m\sqrt{x}$ and see limit depends on $m$. But I am not sure though.
Yes, if you substitute $y=m\sqrt{x}$ you will find a formula that depends on $m$ and the limit along each curve $y=\sqrt{m}$ exists, but you get different values for different $m$, so the limit as $(x,y)\rightarrow (0,0)$ does not exist.