The question is asking to find the limit $\lim_{x \to -1} f(f(x))$ based on the graph provided.

Let's break it down:

1. Step 1: Find $f(-1)$:

   • From the graph, at $x = -1$, the value of $f(x)$ appears to be 3. So, $f(-1) = 3$.

2. Step 2: Find $f(f(-1)) = f(3)$:

   • Now, we need to find $f(3)$. From the graph, at $x = 3$, $f(x)$ is 1. Therefore, $f(3) = 1$.

3. Conclusion: $\lim_{x \to -1} f(f(x)) = f(f(-1)) = f(3) = 1.$

Thus, the answer is $\boxed{1}$.

Would you like more details, or do you have any questions? Here are some follow-up questions:

1. What is the definition of a limit in calculus?
2. How do you evaluate nested functions like $f(f(x))$?
3. What are the key points to observe in a graph when finding limits?
4. Why is continuity important in finding limits?
5. Can the limit exist if there is a discontinuity at the point?

Tip: When evaluating limits involving composite functions, always start by solving the inner function first.