Ques: I want to show that a limit of a function $$f(x,y)=\frac{x^{3}+y^{3}}{x-y}$$ does not exist at point $(0,0)$.
My try: I am just taking path $y=x-x^{3}$ then $$\lim _{(x,y)\rightarrow(0,0)}\frac{x^{3}+y^{3}}{x-y}=\lim_{x\rightarrow 0}\frac{x^{3}+(x-x^{3})^{3}}{x^{3}}=2$$
Then i am taking path $y=2(x-x^{3})$ and i got limit $0$. So, in the both case the limit does not remain same. it means limit of given function does not exist.
Am i right? please give your valuable suggestions!
Best Answer
You don't have to work so hard. For the limit to exist at a point, the function has to be defined in a punctured neighborhood of this point. But your function is undefined whenever $x=y$. Hence the limit does not exist.