https://file.aiquickdraw.com/mathbot/file/1726894235794okhp5qgz.jpg

The problem asks for the limit:

\[
\lim_{x \to -1} f(f(x))
\]

### Step 1: Evaluate \( f(x) \) as \( x \to -1 \)

From the graph, we observe that as \( x \to -1 \), the value of \( f(x) \) approaches 3. Therefore,

\[
f(-1) = 3
\]

### Step 2: Now find \( f(f(x)) \) at \( f(x) = 3 \)

We already know that \( f(-1) = 3 \), so now we need to determine \( f(3) \) from the graph. From the graph, we can see that:

\[
f(3) = 1
\]

### Conclusion:

\[
\lim_{x \to -1} f(f(x)) = f(3) = 1
\]

Thus, the limit is:

\[
\boxed{1}
\]

Let me know if you'd like more details or further explanation. Here are some related questions you can explore:

1. How do you determine the limit from a graph?
2. What are the conditions for a limit to exist at a point?
3. What is the difference between one-sided limits and two-sided limits?
4. How can we apply the chain rule to composite functions like \( f(f(x)) \)?
5. How does continuity affect the calculation of limits?

**Tip:** When evaluating limits from graphs, always check for any discontinuities or jumps at the point you're analyzing.

The graph of the function f is shown. What is lim x -> -1 f(f(x))?

This problem involves finding the limit of a composite function, lim x -> -1 f(f(x)), using the graph of the function. Key concepts include understanding limits, continuity, and how to read graphical information for functions. This problem is suitable for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation