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!