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Let's analyze the problem step by step based on the given information:

### Given Information:
1. \( f \) and \( g \) are continuous functions.
2. \( f(3) = 6 \).
3. \( \lim_{x \to 3} \left[ f(x) + 8g(x) \right] = 19 \).

We are asked to compute:
- (a) \( \lim_{x \to 3} f(x) \)
- (b) \( \lim_{x \to 3} g(x) \)
- (c) \( g(3) \)

### Step 1: Understanding continuity
Since \( f \) and \( g \) are continuous at \( x = 3 \), this implies:
\[
\lim_{x \to 3} f(x) = f(3) = 6
\]
Thus, part (a) is straightforward:

**(a) \( \lim_{x \to 3} f(x) = 6 \)**

### Step 2: Using the given limit
We are given that:
\[
\lim_{x \to 3} \left[ f(x) + 8g(x) \right] = 19
\]
Since limits distribute over addition, we can break this into two parts:
\[
\lim_{x \to 3} f(x) + 8 \lim_{x \to 3} g(x) = 19
\]
Substitute \( \lim_{x \to 3} f(x) = 6 \) into the equation:
\[
6 + 8 \lim_{x \to 3} g(x) = 19
\]
Solve for \( \lim_{x \to 3} g(x) \):
\[
8 \lim_{x \to 3} g(x) = 19 - 6
\]
\[
8 \lim_{x \to 3} g(x) = 13
\]
\[
\lim_{x \to 3} g(x) = \frac{13}{8}
\]

Thus, **(b) \( \lim_{x \to 3} g(x) = \frac{13}{8} \)**.

### Step 3: Continuity of \( g(x) \)
Since \( g(x) \) is continuous, we have:
\[
g(3) = \lim_{x \to 3} g(x)
\]
Thus, **(c) \( g(3) = \frac{13}{8} \)**.

### Final Answers:
- (a) \( \lim_{x \to 3} f(x) = 6 \)
- (b) \( \lim_{x \to 3} g(x) = \frac{13}{8} \)
- (c) \( g(3) = \frac{13}{8} \)

Would you like further details or have any other questions?

Here are five related questions to expand this topic:
1. What are the properties of continuous functions that help solve limits?
2. How can we apply the epsilon-delta definition to prove continuity at a point?
3. Can discontinuous functions have limits that exist?
4. How does continuity of a sum of functions affect their individual limits?
5. How do we interpret limits at infinity for continuous functions?

**Tip**: Always check for continuity when dealing with limits at specific points—it simplifies solving limits.

Suppose that f and g are continuous functions such that f(3) = 6 and lim(x→3) [f(x) + 8g(x)] = 19. Calculate the following:
(a) lim(x→3) f(x)
(b) lim(x→3) g(x)
(c) g(3)

Learn how to solve limit problems involving continuous functions with step-by-step calculations. This problem focuses on continuity, algebraic manipulation, and properties of limits, suitable for students in Grades 11-12.

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