In general I see that we can evaluate $\lim f(g(x)) = f(\lim g(x))$ if $f$ is continuous. How do we know this? How do we prove this?
I tried looking at it with the epsilon-delta definition of a limit but I didn't get very far since I don't know how you even begin to analyze $f(\lim g(x))$.
Assuming that $f$ is continuous at a point $b$, and that $\lim_{x\to a}g(x)=b$, then indeed$$f\left(\lim_{x\to a}g(x)\right)=\lim_{x\to a}f\bigl(g(x)\bigr).\tag1$$In fact, given $\varepsilon>0$, there is a $\delta>0$ such that$$|x-b|<\delta\implies\bigl|f(x)-f(b)\bigr|<\varepsilon.$$And there is a $\delta'>0$ such that$$|x-a|<\delta'\implies\bigl|g(x)-b\bigr|<\delta.$$Putting all this together, we get that$$|x-a|<\delta'\implies\bigl|f\bigl(g(x)\bigr)-f(b)\bigr|<\varepsilon.$$Therefore, $(1)$ holds.