TITLE: Limit composition
QUESTION [9 upvotes]: If I am given the following graph, how do I compute the limit of x->2 of f(f(x))?
I tried to "compose" the limits, but lim x->2 f(x) is 1, but then f is not continuous at 1, so the limit DNE? Is that right?

REPLY [4 votes]: As $x \to 2$ from the left, $f(x) \to 1$ from below; therefore, $$\lim_{x \to 2^-} f(f(x)) = \lim_{x \to 1^-} f(x) = -1$$
But as $x \to 2^+$, we have the exact same behavior: $f(x)$ approaches $1$ from below, so the limit is again $-1$. It's possible to make such limit computations since $f$ is continuous on an interval to the left of $1$.