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 $\boxed{5}$.

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.