How do you decide what two paths to take? For example:

$\displaystyle \frac{x^{2}y}{x^{4}+y^{2}}$

Can I pick any paths I want? For example:

The line $\displaystyle x=0$

$\displaystyle lim_{x\rightarrow 0}$ $\displaystyle \frac{0}{y^{2}}=0$

And the line $\displaystyle y=x^{2}$

$\displaystyle lim_{y\rightarrow x^{2}}$ $\displaystyle \frac{x^{2}*x^{2}}{x^{4}+x^{4}}=\frac{1}{2}$