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Can you graph the function f and then solve the question 

We have successfully graphed the function \( f(x) \), which is a piecewise function defined as:

\[
f(x) =
\begin{cases}
8 - x^2 & \text{for } x < 5 \\
x + 8 & \text{for } x \geq 5
\end{cases}
\]

### Step-by-Step Solution:
We are asked to find the one-sided limit \( \lim_{x \to 5} g(f(x)) \). To proceed, let’s analyze \( f(x) \) as \( x \to 5 \):

- **From the left**: For \( x < 5 \), \( f(x) = 8 - x^2 \). At \( x = 5 \), this yields \( f(5^-) = 8 - 25 = -17 \).
- **From the right**: For \( x \geq 5 \), \( f(x) = x + 8 \). At \( x = 5 \), this gives \( f(5^+) = 5 + 8 = 13 \).

Thus, we need to compute the limits \( \lim_{x \to -17^+} g(x) \) and \( \lim_{x \to 13^-} g(x) \) based on the behavior of \( f(x) \) as \( x \) approaches 5.

Since we are dealing with one-sided limits, it seems that the answer will be linked to options involving \( \lim_{x \to 13^-} g(x) \) or \( \lim_{x \to -17^+} g(x) \). 

The most appropriate option is likely **C: \( \lim_{x \to -17^+} g(x) \)**.

Would you like more details or clarification?

Here are some related questions to expand on this concept:

1. How do you evaluate limits of composite functions like \( g(f(x)) \)?
2. What is the difference between one-sided and two-sided limits?
3. Can you explain the behavior of piecewise functions at their breaking points?
4. What role does continuity play in the evaluation of limits?
5. How can we graphically verify the existence of limits?

**Tip:** Always analyze one-sided limits by checking the behavior from both directions separately!

Can you graph the function f and then solve the question?

Learn how to graph the piecewise function f(x) and solve for the one-sided limit lim_{x→5} g(f(x)). This problem explores key concepts such as piecewise functions, limits, and function composition. Ideal for Grades 11-12 or early college students.

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