What allows us to say that $y=g(x)$?

Question

Suppose we have function $g:A\to B$ and $f:B\to C$. And suppose we have $$\lim_{y\to b} f(y) = f(b)$$ and $$\lim_{x\to a} g(x) = b$$ and $$\lim_{x\to a} f(g(x)) = L,$$ and we need to prove that $$\lim_{x\to a} f(g(x)) =f(\lim_{x\to a} g(x)).$$ While proving it there will be need for stating $y=g(x)$. And usually in the proofs I see people saying "let's set $y=g(x)$". But I do not think that is valid way for justifying it because $x$ and $y$ are bound variables. Am I wrong in my thinking ? Or there is more rigorous way for justifying this ?

Answer 1

Answer-1:

According to the problem it is clear that $f$ is continuous at $x=b$. So continuity of $f$ at $b$ ensures that we can take the limit inside $f$.

Answer-2:

Try $\epsilon-\delta$ definition.

Let, $\epsilon>0$ be given, then $\exists$ a $\delta>0$ s.t. $$|g(x)-g(b)|<\epsilon$$ whenever $|x-b|<\delta$. And again for any $\epsilon'>0$ $\exists$ a $\delta'>0$ s.t. $$|f(x)-f(b)|<\epsilon'$$ whenever $|x-b|<\delta'$. Hence, $$|f(g(x))-f(g(b))|<\epsilon'$$ whenever $|g(x)-g(b)|<\epsilon$ i.e. whenever $|x-b|<\delta_1=\min\{\delta,\delta'\}$.

So, $\displaystyle\lim_{x\rightarrow b}f(g(x))=f(g(b))=f(\lim_{x\rightarrow b}g(x))$ (since it is given that both limits exists) is proved.