Let us use the polar coordinates {x=rcosθy=rsinθ , then the limit (x,y)→(0,0) corresponds to the limit r→0. We have
f(r,θ)=r23r3cos2θsinθ=3rcos2θsinθ
We see that the limit limr→0f(r,θ)=0 exists and independent of θ (as 3cos2θsinθ is a bounded quantity, so limr→0r⋅(bounded quantity)=0)
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