Word Count : 157

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For fixed $x\in\mathbb{R}$, consider $\displaystyle \lim_{y\to 0}\frac{xy}{x^2+y^2}$. If $x\neq 0$, then $x^2+y^2\ge x^2$, hence $$ \left|\frac{xy}{x^2+y^2}\right|\le \frac{|xy|}{x^2}=\frac{|y|}{|x|}\xrightarrow[y\to 0]{}0. $$ If $x=0$, then for all $y\neq 0$, $f(0,y)=0$, so the limit is also $0$. Therefore $$ \lim_{y\to 0}f(x,y)=0\quad\text{for every }x, __________ ____ ________ ____ _______ _________ _______ __________ _____ _______.
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