$$
f(z) = \left\{
 \begin{array}{ll}
 \frac{z^{5}}{|z|^4} & \quad z \neq 0 \\
 0 & \quad z= 0
 \end{array}
 \right.
$$
I am trying to solve this past exam question. Similar question was asked in [Show that $f(z)=\frac{z^5}{|z|^4}$](https://math.stackexchange.com/questions/993739/show-that-fz-fracz5z4) but has not been answered.

**My attempt:**

Let $z=x+iy$ then we can separate the real and imaginary parts of $f(z)$ as

$$
f(z)=u+iv=\left(\frac{x^5-10 x^3 y^2+5 x y^4}{\left(x^2+y^2\right)^2}\right)+i\left(\frac{5 x^4 y-10 x^2
 y^3+y^5}{\left(x^2+y^2\right)^2}\right)
$$ 

$$
\frac{\partial u}{\partial x}=\frac{5 x^4-30 x^2 y^2+5 y^4}{\left(x^2+y^2\right)^2}-\frac{4 x
 \left(x^5-10 x^3 y^2+5 x y^4\right)}{\left(x^2+y^2\right)^3}
$$
$$
\frac{\partial v}{\partial y}=\frac{5 x^4-30
 x^2 y^2+5 y^4}{\left(x^2+y^2\right)^2}-\frac{4 y \left(5 x^4 y-10 x^2
 y^3+y^5\right)}{\left(x^2+y^2\right)^3}
$$
$$
\frac{\partial u}{\partial y}=\frac{20 x y^3-20 x^3 y}{\left(x^2+y^2\right)^2}-\frac{4 y \left(x^5-10
 x^3 y^2+5 x y^4\right)}{\left(x^2+y^2\right)^3}
$$
$$
\frac{\partial v}{\partial x}=\frac{20 x^3 y-20 x
 y^3}{\left(x^2+y^2\right)^2}-\frac{4 x \left(5 x^4 y-10 x^2
 y^3+y^5\right)}{\left(x^2+y^2\right)^3}
$$

I am not sure what to do next. If I substitute $z=0\rightarrow x=0,y=0$ in the equations above then they become infinite because the bottom term $(x^{2}+y^{2})$ becomes zero. So how can I show that $f(z)$ satisfies the Cauchy Riemann equations at $z=0$. Also, how can I show that $f(z)$ is not differentiable at $z=0$?

Thanks.