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Doubt in proof of differentiability of $\begin{equation} f(x,y)= \begin{cases} \frac{x^2y^2}{x^2+y^4}, \neq (0,0)\\ 0 \end{cases}\, \end{equation}$

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Let $f$ be:

\begin{equation}f(x,y)= \begin{cases} \frac{x^2y^2}{x^2+y^4}, (x,y) \neq (0,0) \\ 0, (x,y)=(0,0) \end{cases}\,\end{equation}

Supposedly this is differentiable in (0,0).

But I think if we do the restriction of $f$ to the set $\{(x, \sqrt{x}) : x >0\}$, then the limit of $|\frac{f(x,y)-f(0,0)-0}{\sqrt{x^2+y^2}}|$ = $\frac{|f(x,y)|}{\sqrt{x^2+y^2}}$ as $(x,y)$ goes to $(0,0)$ becomes in module:

$$ \frac{|x|^3}{\sqrt{x^2+x} (2x^2)} =$$

$$ \frac{1}{\sqrt{|x|+1} (2)} =$$

Which goes to $\frac{1}{2}$...

What am I doing wrong? Thank you in advance.


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