TITLE: Problem calculating multivariable limit.
QUESTION [0 upvotes]: Can anyone help me calculate:
$$\lim_{(x,y)\to(0,0)}x\sin\left(\frac{1}{x^2+y^2}\right)$$

REPLY [5 votes]: $\lim\limits_{(x,y) \to (0,0)} \left| x\sin\left(\frac{1}{x^2+y^2}\right)\right |  = \lim \limits_{(x,y) \to (0,0)} | x| \left|\sin\left(\frac{1}{x^2+y^2}\right)\right| \leq \lim \limits_{(x,y) \to (0,0)} | x|=0$
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