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I might just need a hint on where to look for starting on this problem. I can't tell how to prove that this limit does not exist.

If I try $\lim_{(x,0) \to (0,0)} {x\cdot0\cos(0)\over3x^2 + 0^2}$ which is $0$. If I try $\lim_{(0,y) \to (0,0)} {0y\cos(y)\over3(0)^2 + y^2}$, which is still $0$.

Is there another way I can try this to show that the limit does not exist? Thanks.

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Hint:

if you take $x=y$, then you get

$$\lim_{x\to0}\frac{x^2\cos(x)}{3x^2+x^2}=\lim_{x\to0}\frac{\cos(x)}4=\frac14\ne0$$

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    So when you're finding the limit, you can just plug in any arbitrary number? To prove that the limit does not exist all you have to do is show that the limits don't match when approaching from different "directions" right? – 2017-01-24
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    @user406955 Yes. Kinda like those old left and right side limits, but now we can even do $y=f(x)$ or $x=g(y)$ and test those cases. – 2017-01-24