**Get the value of lim┬((x,y)→(0,0))〖xy/√(x^2+y^2 )〗**

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Step 1This is the problem based on multivariate calculus. To find the limit , we can convert given function to the polar from and then we further simplify to find the limit value.Step 2Solufin =\lim _{(x, y) \rightarrow(0,0)} \frac{x y}{\sqrt{x^{2}+y^{2}}}convert to Polar worrinates\begin{array}{l}x=r \cos \theta, y=r \sin \theta \\\lim _{r \rightarrow 0} \frac{r \cos (\theta) \cdot r \sin (\theta)}{\sqrt{r^{2} \cos ^{2} \theta+r^{2} \sin ^{2} \theta}}=\lim _{r \rightarrow 0} \frac{r \cos \theta \cdot r \sin \theta}{r} \\\left(\because \cos ^{2} \theta+\sin ^{2} \theta=1\right) \\=\lim _{\gamma \rightarrow 0} \gamma \cos \theta \sin \theta=\cos \theta \sin \theta \lim _{\gamma \rightarrow 0} \gamma \\=\cos \theta \times \sin \theta \times 0 \\=0 \\\end{array} ...