Let $f(x,y)= \begin{cases} \frac{xy}{x^2 + y^2}, & (x,y) \ne (0,0) \\ 0, & (x,y) = (0,0) \\ \end{cases}$.
a) Show that $\frac {\partial f}{\partial x}|_{(0,0)}$ and $\frac {\partial f}{\partial y}|_{(0,0)}$ exist.
b) Show that $f$ is discontinuous at $(0,0)$.
So I'm having trouble on part (a). Here's what I'm getting: $$\require{cancel}\frac {\partial f}{\partial x}|_{(0,0)} = \lim_{h \to 0} \frac 1 h \left[\frac{(x+h)y}{(x+h)^2 + y^2} - \frac{xy}{x^2 +y^2}\right]_{(0,0)}= \lim_{h \to 0} \left[\frac{y\cancel{h}(y^2 -xh -x^2)}{\cancel{h}(x^2 + y^2)(x^2 +2xh + y^2 +h^2}\right]_{(0,0)}=\left[\frac{y^3 - yx^2}{(x^2 + y^2)^2}\right]_{(0,0)}$$ which is undefined. So it seems like $\frac {\partial f}{\partial x}|_{(0,0)}$ doesn't exist.
By definition we have
$$\frac{\partial f}{\partial x}(0,0)=\lim_{h\to0}\frac{f(h,0)-f(0,0)}{h}=0$$ and the same for $\frac{\partial f}{\partial y}(0,0)$.