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Let's analyze the problem step by step.

We are given:
- \( f(5) = 8 \)
- \( \lim_{x \to 5} \left[ f(x) + 5g(x) \right] = 17 \)

The problem asks us to find the following limits and function values:
(a) \( \lim_{x \to 5} f(x) \)
(b) \( \lim_{x \to 5} g(x) \)
(c) \( g(5) \)

### Step 1: Analyzing \( f(x) \)
Since \( f(x) \) is continuous, the value of the function at a point is equal to the limit at that point. Therefore:
\[
\lim_{x \to 5} f(x) = f(5) = 8
\]
Thus, the answer to part (a) is:
\[
\boxed{8}
\]

### Step 2: Analyzing \( g(x) \)
Next, we know from the given equation that:
\[
\lim_{x \to 5} \left[ f(x) + 5g(x) \right] = 17
\]
Substituting \( \lim_{x \to 5} f(x) = 8 \), we have:
\[
8 + 5 \lim_{x \to 5} g(x) = 17
\]
Solving for \( \lim_{x \to 5} g(x) \):
\[
5 \lim_{x \to 5} g(x) = 17 - 8 = 9
\]
\[
\lim_{x \to 5} g(x) = \frac{9}{5}
\]
Thus, the answer to part (b) is:
\[
\boxed{\frac{9}{5}}
\]

### Step 3: Analyzing \( g(5) \)
Since \( g(x) \) is continuous, the limit and the function value at that point are equal. Therefore:
\[
g(5) = \lim_{x \to 5} g(x) = \frac{9}{5}
\]
Thus, the answer to part (c) is:
\[
\boxed{\frac{9}{5}}
\]

### Summary of answers:
(a) \( \lim_{x \to 5} f(x) = 8 \)  
(b) \( \lim_{x \to 5} g(x) = \frac{9}{5} \)  
(c) \( g(5) = \frac{9}{5} \)

---

Would you like any more details or further clarification?

Here are 5 related questions to deepen your understanding:
1. How do we know that \( f(x) \) and \( g(x) \) are continuous at \( x = 5 \)?
2. What would change if \( f(x) \) were not continuous at \( x = 5 \)?
3. Why do we solve for \( \lim_{x \to 5} g(x) \) by isolating it from the equation?
4. How would you find \( \lim_{x \to 5} \left[ 2f(x) - 3g(x) \right] \) using the values we computed?
5. Can you explain the importance of limits in calculus, particularly for continuous functions?

**Tip:** Continuity is crucial because it guarantees that the value of a function at a point equals the limit at that point, making it easier to calculate limits!

Suppose that f and g are continuous functions such that f(5) = 8 and lim[x->5](f(x) + 5g(x)) = 17. Then we can calculate that (a) lim[x->5] f(x) = ? (b) lim[x->5] g(x) = ? (c) g(5) = ?

Learn how to solve problems involving the limits of continuous functions. This problem focuses on using continuity to calculate limits and function values at specific points. Suitable for Grades 11-12 or introductory college-level calculus students.

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