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The function in the problem is given as:

\[
f(x) = \frac{|9 - 3x|}{x - 3}
\]

The problem asks which statements are true, and these statements involve limits and the value of the function at \(x = 3\). Let's analyze the behavior of the function step by step.

### Step 1: Analyzing the function and its behavior near \(x = 3\)
We need to break down the absolute value:

- For \(x > 3\), \(9 - 3x < 0\), so \(|9 - 3x| = -(9 - 3x) = 3x - 9\).
- For \(x < 3\), \(9 - 3x > 0\), so \(|9 - 3x| = 9 - 3x\).

Thus, we have two cases:
- For \(x > 3\), \(f(x) = \frac{3x - 9}{x - 3} = 3\).
- For \(x < 3\), \(f(x) = \frac{9 - 3x}{x - 3} = -3\).

### Step 2: Evaluating the limit as \(x \to 3\)
- For \(x \to 3^+\), \(f(x) = 3\).
- For \(x \to 3^-\), \(f(x) = -3\).

Since the left-hand limit (\(x \to 3^-\)) and the right-hand limit (\(x \to 3^+\)) are not equal, the limit as \(x \to 3\) does not exist.

### Step 3: Checking \(f(3)\)
At \(x = 3\), the function is undefined because it results in a division by zero. Hence, \(f(3)\) does not exist.

### Step 4: Analyzing the statements
Let's evaluate the given statements based on our analysis:

- **(a)** \(\lim_{x \to 3^+} f(x) = -3\) — False, as \(\lim_{x \to 3^+} f(x) = 3\).
- **(b)** \(\lim_{x \to 0^+} f(x) = 3\) — Irrelevant, as the function does not involve behavior as \(x \to 0\).
- **(c)** \(f(3)\) does not exist — True, since \(f(3)\) results in division by zero.
- **(d)** \(\lim_{x \to 3^-} f(x) = -3\) — True, as \(\lim_{x \to 3^-} f(x) = -3\).
- **(e)** \(\lim_{x \to 3} f(x)\) exists — False, since the limits from the left and right do not match.
- **(f)** \(\lim_{x \to 3} f(x)\) does not exist — True, since the left and right limits are not equal.
- **(g)** \(\lim_{x \to 3} f(x) = 3\) — False, as the limit does not exist.
- **(h)** \(\lim_{x \to 3^+} f(x) = 3\) — True, since \(\lim_{x \to 3^+} f(x) = 3\).

### Final Answer:
The correct statements are **(c)**, **(d)**, **(f)**, and **(h)**.

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Would you like more details on any part of the solution? Here are 5 related questions you can explore:

1. How do you handle absolute value functions when calculating limits?
2. What are the conditions for a limit to exist at a point?
3. Why does division by zero make a function undefined?
4. What is the difference between left-hand and right-hand limits?
5. How does the behavior of a function change around a point of discontinuity?

**Tip:** When evaluating limits, always consider the left-hand and right-hand limits separately to check for discontinuities.

Given the function f(x) = |9 - 3x| / (x - 3), determine which of the following statements about limits and the value of the function at x = 3 are true.

Explore how to calculate the left-hand and right-hand limits of the function f(x) = |9 - 3x| / (x - 3) and understand the conditions for continuity at x = 3. This analysis includes concepts of limits, discontinuity, and behavior of functions near a point of discontinuity.

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