Technically speaking, to show that the limit exists, you need to show that it will always be the same, no matter what path you take.
This can often be a tedious process, so a change in variables is often welcome (e.g. converting to polars)...
This isn't a specific question related to a problem per se, but I'm trying to study for my midterm and I'm having a hard time grasping how you show a limit exists for a function of two or more variables. Disproving it I understand because you just have to find two cases where the limits from different directions don't match, but the definition given by my book on how to calculate whether a limit really does exist is confusing and written in very mathematical terms, which isn't that easy to understand as a beginner. If anyone could clarify the method used to find a limit for a function of two or more variables, that would be extremely helpful!