Word Count : 132

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For fixed $x$ (with $x$ not necessarily zero) evaluate the inner limit $$ \lim_{y\to 0}\frac{xy}{x^2+y^2}=\frac{x\cdot 0}{x^2+0^2}=0. $$ Now take the outer limit: $$ \lim_{x\to 0}\Big(\lim_{y\to 0}\frac{xy}{x^2+y^2}\Big)=\lim_{x\to 0}0=0. $$ Thus $\displaystyle\lim_{x\to0}\lim_{y\to0}f(x,y)=0.$ Similarly, for fixed ___ _______ _____ ____ ______ ____ ___ _______ ________ __________ _______.
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