You could use the squeezing thm on the 3 limits you have to show all are 0.
However showing you have a common limit along 3 different paths does not establish the limit. you would have to prove this for every possible path.
You're method can only be used to show a limit does not exist--i.e. if the limit along different paths is different then the limit does not exist.
However we from your results we suspect the limit is 0
So let's return to the definition of limit. I'll use e for epsilon and d for delta
We need to show
whenever distance[(x,y) to 0] < d
that is whenever x^2 + y^2 < d
then |f(x,y) - L| < e
|f(x,y) - L| = |f(x,y) - 0|
=|(x^2 + y^2)sin1/[x^2+y^2] - 0 | < x^2 + y^2
therefore if delta = e then
|(x^2 + y^2)sin[1/(x^2+y^2)] | < e whenever x^2 + y^2 < d