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**heathrowjohnny** If $\displaystyle f(x,y) = \begin{cases} x \sin \frac{1}{y} \ \ \text{if} \ y \neq 0 \\ 0 \ \ \ \ \ \ \ \ \ \text{if} \ y = 0 \end{cases} $, show that $\displaystyle \lim_{(x,y) \to (0,0)} f(x,y) = 0 $ but that $\displaystyle \lim_{x \to 0} \left(\lim_{y \to 0} f(x,y) \right) \neq \lim_{y \to 0} \left(\lim_{x \to 0} f(x,y) \right). $

So for the first part, I simply plugged in $\displaystyle (x,y) = (0,0) $ and got $\displaystyle 0 $. For the second part, I think the limits oscillate and don't exist?