Questioning 3D Limits: What To Try Next?

[Char. Limit]

So, there was a big long post here with examples and such, but then my internet crashed. So screw the examples, I'll just get straight to my question.

When seeing if a limit exists in 3d (with the limit being (0,0)), I noticed that we often try the lines x=0 and y=0 first. Are these always the best lines to try, or are there others that seem intuitive? If both of these go to the same number, what should I try next?

 

[l'Hôpital]

In general, trying y = mx is always nice. You pick whatever is easy, or whatever looks like it would cause a contradiction.

You can always try polar if it looks like it would work. If it works in polar, it covers all cases.