Under what conditions does the limit f(g(x)) exist?

Under what conditions does

$$ \lim_{x \to a} f(g(x)) $$

exist? I've seen examples like this from Khan that show you can use one-sided limits and discontinuities to break a lot of expectations around limits and still have an answer:

So what's the rule for when this composite limit exists?

Solution 1:

Let $ D_f $ and $ D_g $ be respectively the domains of the functions $ f $ and $ g$.

Let $ B\subset D_f $ and $ A\subset D_g $.

Assume that $ a $ and $ b $ are adherent points of $ A $ and $ B $ .

Then, If

$$\lim_{x\to a,x\in A}g(x)=b$$ $$\lim_{x\to b,x\in B}f(x)=L$$ and $$g(A)\subset B$$ Then $$\lim_{x\to a,x\in A}f(g(x))=L$$ where $ L $ is an adherent point of $ f(B)$.

If you want the left limit at $ a $ and the right limit at $ b $, you will take

$$A=(-\infty,a)\cap D_g$$ $$B=(b,+\infty)\cap D_f$$