Can squeeze theorem be used to prove the nonexistence of a limit?

For squeeze theorem, if $0 \leqslant f(x) \leqslant 1$, can this be used to prove the assertion that the limit $f(x)$ does not exist? I.e. $f(x)$ is $xy/ (x^2+xy+y^2)$, as $(x,y)\rightarrow(0,0)$.

I'm aware you can use the method of approaching from different paths, but was wondering if squeeze theorem was enough? My first guess is obviously this isn't enough to prove anything, since $f(x)$ can still be $0$, but of all the examples I tried, when the squeeze theorem doesn't squeeze $f(x)$ into a number $L$, the limit doesn't exist. Just a coincidence?