Limit of composite piecewise function

Question

Hello,

I was wondering how to solve: $$\lim_{x \to 0} f(f(x))$$ for the attached graph.

This is what I tried: \begin{align} \lim_{x \to 0} f(f(x)) &= f\left[\lim_{x \to 0} f(x)\right] \\ &= f(2) = -1 \end{align} but the correct answer is supposed to be $-2$, how do I arrive at that answer? Thanks

Answer 1

Note that since

$$\lim_{x\to 0} f(x)=2$$

and in a deleted neighborhood of $x=2$ we have $f(x)<2$ then

$$\lim_{x\to 0} f(f(x))=\lim_{y\to 2^-} f(y)=-2$$

Answer 2

Since $f$ is continuous at $x=0$ you have $\lim_{x \to 0} f(x) = 2$. Furthermore, since $f(x) \le $ for all $|x| <1$ we see that as $x \to 0$ (but not equal to) we also have $f(x) < 2$.

Hence as $x \to 0$ we have $f(x) \uparrow 2$ (that is, less than as it approaches).

From the graph, we see that for $y \in (1,2)$ we have $f(y) = 2(1-y)$, hence we see that $f(f(x)) \to \lim_{y \uparrow 2} f(y) = -2$.