How do I show that the limit $$\lim_{(x,y)\to(0,0)}\frac{1}{3}x^{-2/3}\cos(y)$$ does not exist?
I have to prove that the function $$f(x,y)=x^{1/3}\cos(y)$$ is not differentiable in $(0,0)$, so I calculated (using limit): $$\frac{\partial f}{\partial x}(0,0)=0$$ And now I'm trying to show that $\dfrac{\partial f}{\partial x}$ is not continuous in $(0,0)$, but I can't answer my question (the limit above).
We have that $f(x,0)=x^{1/3}.$ Thus
$$\frac{\partial f}{\partial x}(0,0)=\lim_{x\to 0}\frac{f(x,0)-f(0,0)}{x}=\lim_{x\to 0}\frac{x^{1/3}}{x}=\lim_{x\to 0} x^{-2/3}.$$ Are you able to show that this limit doesn't exist?