https://file.aiquickdraw.com/mathbot/file/1726117996792ixr2pya8.jpg

Let's focus on solving **Question 4** based on the given image.

### Question 4:
Let \( p(x) = g(x) - f(x) \). Is \( p(x) \) continuous at \( x = 4 \)? Justify your answer.

### Solution:

To determine whether \( p(x) = g(x) - f(x) \) is continuous at \( x = 4 \), we need to check three conditions for continuity:

1. **Existence of \( p(4) \)**: Check if \( p(4) \) exists by calculating the value of \( g(4) - f(4) \).
2. **Existence of the limit**: Check if \( \lim_{x \to 4} p(x) \) exists, i.e., calculate the limits \( \lim_{x \to 4^-} p(x) \) and \( \lim_{x \to 4^+} p(x) \).
3. **Equality of the limit and function value**: Verify if \( \lim_{x \to 4} p(x) = p(4) \).

### Step 1: Compute \( p(4) \)
From the graphs:
- \( f(4) = 2 \)
- \( g(4) = 0 \)

So,
\[
p(4) = g(4) - f(4) = 0 - 2 = -2
\]

### Step 2: Check the limit \( \lim_{x \to 4} p(x) \)

We need to check both one-sided limits:

- **Left-hand limit (\( x \to 4^- \))**:
  - As \( x \) approaches 4 from the left, \( f(x) \to 1 \) and \( g(x) \to 0 \).
  - So, \( p(x) = g(x) - f(x) \to 0 - 1 = -1 \).

- **Right-hand limit (\( x \to 4^+ \))**:
  - As \( x \) approaches 4 from the right, \( f(x) \to 2 \) and \( g(x) \to 0 \).
  - So, \( p(x) = g(x) - f(x) \to 0 - 2 = -2 \).

Since the left-hand limit and right-hand limit are not equal (\( -1 \neq -2 \)), the limit \( \lim_{x \to 4} p(x) \) **does not exist**.

### Step 3: Conclusion
Since the limit \( \lim_{x \to 4} p(x) \) does not exist, \( p(x) \) is **not continuous** at \( x = 4 \).

### Final Answer:
\( p(x) \) is **not continuous** at \( x = 4 \), because the left-hand and right-hand limits do not match.

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Would you like more details on any of the steps or any additional questions?

Here are 5 related questions:
1. What are the conditions for a function to be continuous at a point?
2. How do you evaluate limits graphically?
3. What is the difference between one-sided and two-sided limits?
4. How do you determine the continuity of piecewise functions?
5. How do function operations like addition or subtraction affect continuity?

**Tip**: When checking continuity, always ensure that both the limit and function value exist, and that they are equal!

Let p(x) = g(x) - f(x). Is p(x) continuous at x = 4? Justify your answer.

Learn how to determine the continuity of the function p(x) = g(x) - f(x) at x = 4. This problem involves verifying the conditions for continuity, calculating limits, and understanding function operations. Suitable for high school calculus students.

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