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We know by the composition of limits that if $\lim_{x \to c}g(x)=b$ and $\lim_{x \to b}f(x)=L$ and $g(x) \neq b$ in some neighborhood of $c$ then $$\lim_{x \to c}f(g(x))=L$$ But If we choose a function $f(x)$ which is defined on a set $X$ in such a way that every neighbourhood of $b$ contains some elements of the set $X$ and $\lim_{x \to b}f(x)=L$ and suppose we choose a function $g(x)$ such that $g:R \to Y$ and $X∩Y = \phi $ and also $\lim_{x \to c}g(x)=b$ then we see that $f(g(x))$ is not defined at any point and so the limit $$\lim_{x \to c}f(g(x))$$. So how to deal with composition of limits in such cases??

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Well, $f \circ g$ is only defined on $X \cap Y$ and that is empty, but the definition of limit requires for $h(x)=f(g(x))$ to be defined in a neighbourhood of $c$, so there is no limit...

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