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.