TITLE: A limit that involves two variables
QUESTION [7 upvotes]: I'm trying to compute this limit $\lim_{(x,y) \to (0,0)}2x\sin^2(\frac{1}{y})$, but WolframAlpha says that it does not exist. 
I'm not quite sure why. I do understand that there are oscillations coming from the $\sin(1/y)$. However, $x \to 0$ as well. Shouldn't that crush the function to zero?
Also, I know that $\lim_{x \to 0} x \sin(\frac{1}{x}) = 0$. Isn't that pretty much the same idea as the limit in question?

REPLY [2 votes]: $$|2x\sin^2(1/y)|\le 2|x|\cdot |1|\le 2|x|.$$
Since $2|x|\to 0$ as $x\to 0$, the limit is not undefined.