# Composition of Limits In The Most General Case

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??

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...