From this video, I've learned to expect if a limit exists or not is by comparing the combined power on the numerator is bigger than than the combined power on the denominator.
For example, when considering limit as $(x,y) \rightarrow (0,0)$the rational function $$\frac{3x^2y}{x^2 + y^2}$$
The numerator has an exponential power of three ($x^2 * y)$ whereas the denominator has an exponential power of only 2.
However, I tried to use this intuition to surmise the existence of a limit as$(x,y) \rightarrow (0,0)$ for the rational function $\frac{x^4 - 4y^2}{x^2 + 2y^2}$.
As it has an exponential power of 4 on the numerator and an exponential power of 2 on the denominator, I expected the limit to exist. After struggling to prove this using the squeeze theorem, I reassessed using the path test, which quickly proved that it does not.
What are ways of quickly assessing whether we can expect a limit to exist?
Using polar coordinates $$x=r\cos (t) \;\;\; y=r\sin (t) $$
the multivariate function becomes
$$f (x,y)=g (r,t)=3r\sin (t)\cos (t) $$
as $r\to 0$, $g (r,t) $ goes to zero for all $t $. So the limit exists and is zero.