https://file.aiquickdraw.com/mathbot/file/1726046750453q74yc7eb.jpg

https://file.aiquickdraw.com/mathbot/file/1726046761700f63dctrl.jpg

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:** \( \lim_{x \to 3} f(g(x)) = 0 \).

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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?

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**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!

Use the graph to determine the following limits and values:
(a) lim_{x → 3^-} g(x)
(b) lim_{x → 3^+} g(x)
(c) lim_{x → 3} g(x)
(d) g(-3)
(e) lim_{x → 0} (2f(x) + g(x))
(f) lim_{x → 3} f(g(x))

Analyze the graph to determine various limits and function values, focusing on left-hand and right-hand limits, continuity, and composition of functions. Suitable for Grades 10-12 students learning calculus concepts.

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