Let $u=f(z)$ be an elementary function and $z=g(x,y)$ a two-variable elementary function. Suppose we have $g(x,y)$ is continuous at point $(a,b)$, $g(a,b)=c$, and $\lim_{z\to c}f(z)=A$. When is it true that $\lim_{(x,y)\to(a,b)}f(g(x,y))=A$? If it's not true in general, are there any counter examples?
Composition of two-variable limits
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limits