Now I don't understand what your picture is trying to tell me either. The proof of the limit in your first post is just fine.
You have now posted 3 different pictures that appear to involve three different problems. As Dick told you before, the first is correct. This third one is also correct- except for spelling: since you get two different limits by approaching (0,0) along two different paths, the limit does not exist.
My answer of the first question :
is correct or a mistake?
I'm not sure what the second picture was supposed to represent. But one thing you should be careful about: The line y= 2x, for example, in R2, is not in the complex plane.