When taking the limit of function composition, what is the best way to write the intermediate step?
Suppose we have two functions $f(x)$ and $g(x)$ that are continuous where $\lim_{x\to a} g(x)=b$ and $\lim_{x \to b} f(x)=c$ and we are trying to evaluate $\lim_{x \to a}f(g(x))$, how do we write out the intermediate step in the following evaluation?
$\lim_{x \to a}f(g(x))=\lim_{x \to b} f(x)=c$
or
$\lim_{x \to a}f(g(x))=\lim_{g(x) \to b} f(g(x))=c$
The first is uncomfortable because the $x$ variables are not referring to the same variable, but the second seems unusual with $g(x) \to b$ being written under the limit.
Or is there a third way that is still better notation?
In general both notation are fine, I would prefer the following
$$\lim_{x \to a}f(g(x))=\lim_{y \to b} f(y)=c$$
since $f$ is continuous we can also use the following
$$\lim_{x \to a}f(g(x))=f\left(\lim_{x \to a} g(x)\right)=f(b)=c$$