I understand what you mean but, for example, if I take the fourth limit:
lim(x,y)->(0,0) (x^2 * y^2) / (x^4 + 5y^4)
If I evaluate this when x=y then I end up with (x^4)/(6x^4) which gives a limit of 1/6.
However, if I evaluate this at (x,0) which should also work, I get a limit of 0/(x^4) =...