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).

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.