https://file.aiquickdraw.com/mathbot/file/1730865613872fbajnoat.jpg

To analyze this question, let's go through each of the given statements concerning the piecewise-linear function \( f(x) \):

The function is defined as:
\[
f(x) = \begin{cases} 
2x - 2 & \text{for } x < 3 \\
2x - 4 & \text{for } x \geq 3 
\end{cases}
\]

### Statement I: \(\lim_{x \to 3} f(x)\) does not exist.
To check if the limit exists, we need to calculate the left-hand limit (\(x \to 3^-\)) and the right-hand limit (\(x \to 3^+\)).

1. **Left-hand limit** (\(x \to 3^-\)):
   \[
   \lim_{x \to 3^-} f(x) = \lim_{x \to 3^-} (2x - 2) = 2(3) - 2 = 4
   \]

2. **Right-hand limit** (\(x \to 3^+\)):
   \[
   \lim_{x \to 3^+} f(x) = \lim_{x \to 3^+} (2x - 4) = 2(3) - 4 = 2
   \]

Since the left-hand limit (4) does not equal the right-hand limit (2), the limit \(\lim_{x \to 3} f(x)\) does not exist. **Statement I is true.**

### Statement II: \(\lim_{x \to 3^+} \frac{f(x) - f(3)}{x - 3} = 2\)
To evaluate this, we need to check the derivative from the right at \(x = 3\), which involves calculating the expression:
\[
\lim_{x \to 3^+} \frac{f(x) - f(3)}{x - 3}
\]

First, find \(f(3)\):
\[
f(3) = 2(3) - 4 = 2
\]

Then, using the definition of the derivative from the right,
\[
\lim_{x \to 3^+} \frac{f(x) - 2}{x - 3} = \lim_{x \to 3^+} \frac{(2x - 4) - 2}{x - 3} = \lim_{x \to 3^+} \frac{2x - 6}{x - 3} = \lim_{x \to 3^+} 2 = 2
\]
This limit equals 2, so **Statement II is true**.

### Statement III: \(f'(3) = 2\)
For \(f'(3)\) to exist, \(f(x)\) would need to be differentiable at \(x = 3\). However, differentiability requires that the left-hand and right-hand derivatives at \(x = 3\) are equal.

1. **Left-hand derivative** (\(x < 3\)):
   \[
   \frac{d}{dx}(2x - 2) = 2
   \]

2. **Right-hand derivative** (\(x \geq 3\)):
   \[
   \frac{d}{dx}(2x - 4) = 2
   \]

Since both the left-hand and right-hand derivatives are equal to 2, \(f'(3) = 2\). **Statement III is true**.

### Conclusion
Since all three statements (I, II, and III) are true, the correct answer is **(C) I, II, and III**.

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Would you like a deeper explanation on any part of this solution?

Here are some related questions to consider:
1. What conditions must a piecewise function meet to be differentiable at a point?
2. How do we determine the continuity of a piecewise function?
3. What role do left-hand and right-hand limits play in verifying limits at specific points?
4. How can we compute derivatives for different segments of a piecewise function?
5. How do we evaluate whether a piecewise-defined limit exists at a boundary?

**Tip:** For piecewise functions, always check both left-hand and right-hand limits and derivatives separately at the boundary points.

Let f be the piecewise-linear function defined below. Which of the following statements are true?

f(x) = { 2x - 2 for x < 3, 2x - 4 for x ≥ 3 }

I. lim(x→3) f(x) does not exist.
II. lim(x→3^+) (f(x) - f(3))/(x - 3) = 2
III. f'(3) = 2

(A) II only
(B) I and II only
(C) I, II, and III
(D) None

Explore the conditions for limit existence, continuity, and differentiability in a piecewise-defined function. This problem examines left-hand and right-hand limits at x=3, checks the existence of the limit, and evaluates the derivative of the function at a boundary point. Suitable for high school students studying calculus.

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