Show that the limit (xy+y^3)/(x^2+y^2) as (x y) tends to (0 0) does not exist

Question : Show that the limit (xy+y^3)/(x^2+y^2) as (x y)->(0 0) does not exist \[limit_{\left(x,y\right)\rightarrow(0,0)}\frac{xy+y^3}{x^2+y^2} does \ not \ exist \]Solution :Let \[ L=limit_{\left(x,y\right)\rightarrow(0,0)}\frac{xy+y^3}{x^2+y^2}\] Limit along the path y=x \[L=limit_{x\rightarrow 0}\frac{x^2+x^3}{x^2+x^2}\] \[L=limit_{x\rightarrow 0}\frac{x^2(1+x^2)}{2x^2}\] \[L=limit_{x\rightarrow 0}\frac{1+x^2}{2}\] \[L=\frac{1+0}{2}\] \[L=\frac{1}{2}\] limit along the…

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