I have been struggling with how to choose which two paths to prove the multivariable limit does not exist.
For example, enter image description here $$ \lim_{(x,y) \to (0,0)} \dfrac{x^2y}{x^2+y^2}$$
I tried different paths, $y=x, y=x^2, y=x^3, y=x^2-x, y=x^3-x, \cdots$. They all give me that the limit is equal to $0$, but the limit actually does not exist. How can I choose the correct paths and is there any strategy for choosing the paths to prove the limit does not exist? Thank you so much!