Use the definition of partial derivative to show that ∂f/ ∂x (0, 0) does not exist

Question

[I just need help with part b of this problem][1]

Let $f:\mathbb{R}^2\to\mathbb{R}^2$ be defined by $f(0,0)=0$ and $f(x,y)=\frac{2x^2-y^2}{x^2+y^2}$ if $(x,y)\neq (0,0)$.

a) Show that $f$ is not continuous at $(0,0)$.

b) Use the definition of partial derivative to show that $\frac{\partial f}{\partial x}(0,0)$ does not exist.

Answer 1

Hint: You just have to setup the proposed limit $$\lim_{h \to 0} \frac{f(0+h,0)-f(0,0)}{h} = \lim_{h \to 0}\frac{2-0}{h} = \lim_{h \to 0}\frac{2}{h}.$$What about lateral limits?