In general, is there an intutive way to "know" when a multivariate limit exists/does not exist? I know that I can use trial-and-error with some paths of approach to show that it does not exist, but often this takes significant time, and is not that feasible under exam conditions.
For e.g., if this were a rational expression (i.e., ignore the log term) of polynomials in both the denominator and numerator, then I could compare degrees, and have an "intuition" that the bottom degree of 6 will "shrink faster" than the top, causing me to think that the limit does not exist.
Can I generalise this intution to non-rational functions? i.e., a very hairy composition of several functions in both the denominator and numerator. Or perhaps a function that's not even in the form of a fraction.