your method is sound though. note that if we approach (0,0) along x = 2y we get -2/3 as the limit. this means the limit doesn't existAs both limits are different, the limit doesn't exist.
However I'm unsure I didn't make an error by taking for the second limit. Since when , this doesn't even make sense to take such an .
The domain of is any such that .
(the curve you approach the origin must actually go through the origin. the line x = 1 does not.)