to use the precise definition of the limit (and i am not totally sure what you are referring to when you say this, but) you have to have an idea of what the limit is to begin with. so you approach the limit along several lines, and you should always get two, so you would believe 2 is the limit and then try to prove it by the definition of the limit.

here's another way though: switch to polar coordinates

recall that, in polar coordinates:

so as , we have

thus,

so we used a change of variable to change to a limit in one variable, i'm sure you can take it from here