CONTEXT: University first year multivariate calculus maths course.
I need to find the limit of $f:ℝ^2$\ $(0,0) \to ℝ$, or show that it does not exist.
$$f(x,y)=\frac{y^2x}{y^4+2x^2}$$
I know that it does not exist, but don't know how to show this (without writing a rigorous proof as that is beyond the scope of the course) by finding two different limits along two different paths within the domain.
I have checked along these paths: $x$-axis, $y$-axis, $y=x$ and so far they all give limits of $0$. I also looked along the path $x=y^2$ which gave a limit of $\frac{1}{3}$, but was unsure of whether you can consider nonlinear paths (can I use this?).
Any guidance would be greatly appreciated!
Best Answer
You are right.
We have $f(x,0)=0$ for $x \ne 0$, hence $f(x,0) \to 0$ as $x \to 0.$
Furthermore $f(y^2,y)=1/3$ for $y \ne 0$, hence $f(y^2,y) \to 1/3$ as $y \to 0.$
Conclusion: $\lim_{(x,y) \to (0,0)}f(x,y)$ does not exist.