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.