What is the limit of xy2(x2+y2)\frac{xy^2}{(x^2+y^2)} as (x,y)(x,y) approaches (0,0)(0,0)? Thanks for the help in advance.

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2 Answers

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Verify the inequality

|x|y2x2+y2≤|x|\frac{|x|y^2}{x^2+y^2} \le |x|

For x=0x=0 you have f(0,y)=0y20+y2=0f(0,y) = \frac {0y^2}{0+y^2} = 0,

and for y=kxy=kx it is f(x,y)=k2x3(1+k2)x2=k21+k2xf(x,y) = \frac {k^2x^3}{(1+k^2)x^2} = \frac {k^2}{1+k^2}x.

What are possible values of k21+k2\frac {k^2}{1+k^2}…?

And what are possible values of xx inside a circle of radius rr centered at (0,0)(0,0)…?