We use the definition "with and " to obtain the rules about limits and sum or product.
In cases like that, you can use these results instead of the " ".
You can, for example, replace x with x'= x- 1 so x= x'+1 and y with y'= y- 1so y= y'+1. Then, the problem becomes to prove that goes to 3 as x' and y' go to 0. Now change to polar coordinates: and . Now we have . The reason I "shifted" the point to (0, 0) is that r alone measures the distance to the origin. Now, as r goes to 0, both and go to 0 as r goes to 0, no matter what is and so the limit is 3.
And, now, I think it would not be too difficult to get a proof from that. Given any epsilon, we want so that if then
Find an upper bound on and take to be less than divided by that upper bound.