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Here is a problem from an old comprehensive exam that I am trying to solve

Problem: let $f:\mathbb{R}^{2} \to \mathbb{R}$ be a function defined as follows:

$f\left ( x,y \right )=\frac{\left ( x^{2}-y \right ).y^{2}}{x^{4}+y^{2}}$ if $\left ( x,y \right )\neq \left ( 0,0 \right )$

$f\left ( x,y \right )=0$ if $\left ( x,y \right )=\left ( 0,0 \right )$

The question is : Investigate the differentiability of $f$ at the point $\left ( 0,0 \right )$

Here is what I did so far: $\frac{\partial f}{\partial x}\left ( 0,0 \right )=0 $ and $\frac{\partial f}{\partial y}\left ( 0,0 \right )=-1 $

Now I applied the condition for differentiability for $f$ at the point $\left ( 0,0 \right )$:

$lim_{\left ( x,y \right ) \to \left ( 0,0 \right )}\frac{\left \| f\left ( x,y \right )-f\left ( 0,0 \right )-\bigtriangledown f\left ( 0,0 \right ).\left ( x,y \right ) \right \|}{\sqrt{x^{2}+y^{2}}}$ has to be $0$ if $f$ is differentiable at the desired point.

After simplifying the above limit, I got the following limit:

$lim_{\left ( x,y \right ) \to \left ( 0,0 \right ) }\frac{x^{2}y^{2}+x^{4}y}{\left ( x^{2}+y^{4} \right ).\sqrt{x^{2}+y^{2}}}$ . Here is where I am stuck. I cannot evaluate this limit. I tried everything, like evaluating the limit through different paths... but nothing seems to work out for me.

Any help please on how to finish my proof?

2 Answers 2

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$\bf Hint:$ Using polar coordinates the equation turns into: $lim_{r\to 0}\frac{r^4cos^2\theta sin^2\theta+r^5cos^4\theta sin\theta}{r^3cos^2\theta+r^5sin^4\theta} $

Edit: Note that if $sin\theta=0$ then the equation turns into $lim_{r\to 0}\frac{r^4+r^5}{r^3}=lim_{r\to 0}r+r^2=0$.

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    Yes, the limit does not exists in that case.2012-02-04
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You can separate the limit (that you got in the definition of diferentiability) as the limit of a sum. Each sumand : you can express as the products of bounded functions multiplied by functions that converge to zero: X^2/x^2+y^4 is bounded, y/square root of x^2+y^2 is bounded. You know that a bounded function multiplied by a factor that converges to 0 , gives a product that converges to zero.