https://file.aiquickdraw.com/mathbot/file/1726970946663d0gaq7yh.jpg

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.

The graph of the function f is shown above. What is \( \lim_{x \to -1} f(f(x)) \)?

Solve the limit problem \( \lim_{x \to -1} f(f(x)) \) using graphical analysis of the function. This problem explores key calculus concepts such as composite functions, limits, and continuity. Suitable for students in Grades 11-12 studying pre-calculus or calculus.

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