https://file.aiquickdraw.com/mathbot/file/17268540809004rkvq315.jpg

The problem in the image provides the following information:

- The graph of \( f(x) \) is shown.
- You are given:
  - \( \lim_{x \to 1^-} g(x) = -3 \)
  - \( \lim_{x \to 1^+} g(x) = 4 \)
  - \( g(1) = 5 \)

The task is to find \( \lim_{x \to 6} g(f(x)) \).

### Steps to Solve:

1. **Identify \( f(6) \) from the graph of \( f(x) \):**
   - Looking at the graph, observe the value of \( f(x) \) when \( x = 6 \).
   - The graph shows that \( f(6) = 1 \) (since the point at \( x = 6 \) corresponds to \( y = 1 \)).

2. **Substitute \( f(6) \) into \( g(f(x)) \):**
   - Since \( f(6) = 1 \), we need to find \( \lim_{x \to 6} g(f(x)) = g(1) \).

3. **Use the given information about \( g(1) \):**
   - We know that \( g(1) = 5 \).

Thus, \( \lim_{x \to 6} g(f(x)) = 5 \).

### Final Answer:
The limit is \( \boxed{5} \).

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Would you like further clarification on any step?

Here are 5 related questions to explore further:
1. How do you compute a limit involving composite functions like \( g(f(x)) \)?
2. Why is the left-hand limit and right-hand limit for \( g(x) \) important in this problem?
3. How do you determine if a function is continuous from a graph?
4. What role does the graph of \( f(x) \) play in finding \( f(6) \)?
5. How would the solution change if \( g(1) \) were undefined or discontinuous?

**Tip:** Always examine whether the functions involved are continuous at the point you are evaluating, as limits involving discontinuities require special attention.

If \( \lim_{x \to 1^-} g(x) = -3 \), \( \lim_{x \to 1^+} g(x) = 4 \), and \( g(1) = 5 \), find \( \lim_{x \to 6} g(f(x)) \) based on the graph of \( f(x) \).

Learn how to find the limit \( \lim_{x \to 6} g(f(x)) \) using the graph of the function f(x) and the given limit information for g(x). This problem explores key calculus concepts such as composite functions, continuity, and limits. Suitable for students in Grades 11-12.

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