Let's begin by analyzing the information given in the picture and solve each question step by step.

Graph Observations:

• The red curve is $f(x)$.
• The blue curve is $g(x)$.
• Solid dots indicate the values where the function is defined, while hollow dots represent holes (undefined values at those points).
• The axes help us approximate the behavior of the functions around the specified points.

Solving the Problems:

(a) $\lim_{x \to 3^-} g(x)$

We are looking for the left-hand limit of $g(x)$ as $x$ approaches 3.

• As $x \to 3^-$ (from the left), the blue curve appears to approach a value slightly below $0$.
• Answer: $\lim_{x \to 3^-} g(x) = -1$.

(b) $\lim_{x \to 3^+} g(x)$

We are now looking for the right-hand limit of $g(x)$ as $x$ approaches 3.

• As $x \to 3^+$ (from the right), the blue curve approaches $-1$.
• Answer: $\lim_{x \to 3^+} g(x) = -1$.

(c) $\lim_{x \to 3} g(x)$

The overall limit at $x = 3$ exists if both the left-hand and right-hand limits are equal.

• Since both the left-hand and right-hand limits equal $-1$, the limit exists and is $-1$.
• Answer: $\lim_{x \to 3} g(x) = -1$.

(d) $g(-3)$

We need to find the value of $g(x)$ at $x = -3$.

• At $x = -3$, the blue curve has a solid dot at $y = -1$, indicating that the value of $g(-3)$ is $-1$.
• Answer: $g(-3) = -1$.

(e) $\lim_{x \to 0} (2f(x) + g(x))$

We need to compute the limit of the expression $2f(x) + g(x)$ as $x \to 0$.

• From the graph:
  • As $x \to 0$, $f(x) \to 0$ (since the red curve touches the origin).
  • As $x \to 0$, $g(x) \to 1$ (from the blue curve).
• Now, using these limits: $\lim_{x \to 0} (2f(x) + g(x)) = 2 \times 0 + 1 = 1$
• Answer: $\lim_{x \to 0} (2f(x) + g(x)) = 1$.

(f) $\lim_{x \to 3} f(g(x))$

We need to find the limit of $f(g(x))$ as $x \to 3$.

• From part (c), we know that $\lim_{x \to 3} g(x) = -1$.
• Now we need to evaluate $f(x)$ as $x \to -1$:
  • From the graph, $f(-1) = 0$ (since the red curve touches the x-axis at $x = -1$).
• Therefore: $\lim_{x \to 3} f(g(x)) = f(\lim_{x \to 3} g(x)) = f(-1) = 0$
• Answer: $\lim_{x \to 3} f(g(x)) = 0$.

Let me know if you need more details on any of the solutions! Here are 5 related questions that can extend the learning:

1. How would the limit $\lim_{x \to 0} f(x)$ differ if there was a hole at $x = 0$ in $f(x)$?
2. What is the significance of left-hand and right-hand limits when analyzing a piecewise function?
3. Can you find $\lim_{x \to -3^+} g(x)$ using the graph?
4. How does the composition of functions, like $f(g(x))$, behave when the inner function has discontinuities?
5. How would you use algebraic techniques to confirm a limit value if the graph was not available?

Tip: When calculating limits, always check if the limit exists from both sides (left-hand and right-hand). If they differ, the limit does not exist at that point!