So a recent post gave a nasty 2-variable function:
$$f(x,y) = x^2y/(x^4+y^2)$$
and after changing to polar coordinates, you get that the limit is always equal to zero if you hold $\theta$ fixed and take $r \to 0$. However if you put $\sin \theta = r$ then you get that the limit is $1/2$ as $r \to 0$, so the limit as $(x,y) \to (0,0)$ doesn't exist. So that got me wondering, are there any non-trivial conditions on a function that guarantee you can just hold $\theta$ fixed and let $r \to 0$ to verify whether a limit exists, yet some limits may not exist?
If you have a rational function where the numerator and denominator are homogeneous polynomials in two variables and have the same degree, you can always rewrite the function in one parameter (the slope of the line) using the substitution $y=tx$. This works since we have a function defined on lines through the origin, as $f(\lambda x,\lambda y)=f(x,y)$ $\forall \lambda \in \mathbb{R}-\{0\}$. For example:
$\frac{x^2y^2+3x^3y}{x^4+xy^3}=\frac{x^2(t^2x^2)+3x^3(tx)}{x^4+x(t^3x^3)}$=$\frac{t^2+3t}{t^3+1}$
In the above the limit doesn't exist, since choosing different $t$ give different (constant) values of the function along that line. The limit will then only exist in the case of a constant function. So I guess this would be one sufficient condition. If you want something which is also necessary, I'm not really sure.