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Consider the function $$f(x,y)=\begin{cases}(x^2+y^2)\sin\left(\frac1{\sqrt{x^2+y^2}}\right)&\text{if }(x,y)\ne(0,0)\\0&\text{if }(x,y)=(0,0).\end{cases}$$ Show that the partial derivatives $f_x(0,0)$ and $f_y(0,0)$ exist.

What they are asking here?

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They want you to show that $f_x(0,0)$ and $f_y(0,0)$ exist.

In particular $\lim_{h\to 0} \frac{f(h,0)-f(0,0)}{h} = f_x(0,0)$ and $\lim_{h\to 0} \frac{f(0,h)-f(0,0)}{h}=f_y(0,0)$ exist.

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