# Multivariable limit: Proving Limit does not exist, how to choose the best path

I am given the problem $$\lim_{(x,y) \to (0,0)}\frac{ y\sin^2{x}}{x^4 + y^2}$$ I know the first path to choose is either the x-axis or y-axis. This gives the limit of 0. Now I don't know how to choose the second path to prove that it does not exist.

Please forgive me for the format, I am new and I haven't learned it yet. Also sorry for my English if it's not properly correct.

• The answers give examples of paths that do this. As for how to find such examples, typically one first tries linear approaches, namely $y = ax$ where $a$ is a constant, and see if the expression is constant that varies with $a.$ If this doesn't work (i.e. you get a constant value for the limit that doesn't vary with $a),$ then try quadratic approaches, namely $y = x^2$ (or more generally, $y = ax^2$ where $a$ is a constant), or cubic approaches, or $n$'th degree approaches (i.e. $y = ax^n$ where both $a$ and $n$ are constants). (continued) – Dave L. Renfro Aug 19 '19 at 10:44
• Here is an example where you need to go beyond "polynomial approaches". – Dave L. Renfro Aug 19 '19 at 10:45

Hint: Take $$y=x^{2}$$ and show that the limit is $$\frac 1 2$$. Use the fact that $$\frac {\sin \, x } x \to 1$$ as $$x \to 0$$.
If $$f(x,y)=\frac{y \sin^2 x}{x^4+y^2}$$, then compute
$$\lim_{x \to 0}f(x,0)$$ and $$\lim_{x \to 0}f(x,x^2)$$.