The given function is $f(x) = \frac{|2x - 4|}{2 - x}$, and the question asks which statements about this function are correct.

Let's analyze each statement one by one:

1. $f(2)$: $f(2) = \frac{|2(2) - 4|}{2 - 2} = \frac{|4 - 4|}{0} = \frac{0}{0} \quad \text{(undefined)}$ So, the value of $f(2)$ does not exist.

2. Limits as $x \to 2$:

   The behavior of the function around $x = 2$ needs to be checked by approaching from the left and right:

   • For $x \to 2^+$ (approaching 2 from the right):

     • When $x > 2$, $|2x - 4| = 2x - 4$, so the function becomes: $f(x) = \frac{2x - 4}{2 - x}$ As $x \to 2^+$, the denominator $2 - x$ approaches 0 from the negative side, and the numerator $2x - 4$ approaches 0 from the positive side. Therefore: $\lim_{x \to 2^+} f(x) = -\infty$

   • For $x \to 2^-$ (approaching 2 from the left):

     • When $x < 2$, $|2x - 4| = -(2x - 4) = 4 - 2x$, so the function becomes: $f(x) = \frac{4 - 2x}{2 - x}$ As $x \to 2^-$, the denominator $2 - x$ approaches 0 from the positive side, and the numerator $4 - 2x$ approaches 0 from the positive side. Therefore: $\lim_{x \to 2^-} f(x) = +\infty$

   Since the one-sided limits do not match, the two-sided limit does not exist.

3. Other limits and values:

   • $\lim_{x \to 0^+} f(x)$: $f(x) = \frac{|2x - 4|}{2 - x}$ As $x \to 0^+$, the numerator approaches 4 and the denominator approaches 2, so: $\lim_{x \to 0^+} f(x) = \frac{4}{2} = 2$

Correct Answers:

• (a) $f(2)$ does not exist.
• (d) $\lim_{x \to 2} f(x)$ does not exist.
• (g) $\lim_{x \to 2} f(x)$ exists (this statement is false).
• (h) $\lim_{x \to 0^+} f(x) = 2$.

Would you like more details or explanations about any part of the solution?

Here are 5 follow-up questions to deepen understanding:

1. Why does $f(2)$ result in an undefined form $\frac{0}{0}$?
2. How does the behavior of $f(x)$ change when $x \to 2$ from the left and right?
3. What is the significance of checking one-sided limits in this problem?
4. Why does $\lim_{x \to 2^+} f(x) = -\infty$ but $\lim_{x \to 2^-} f(x) = +\infty$?
5. How can we verify limits analytically using the piecewise nature of absolute values?

Tip: Always evaluate one-sided limits separately when there is an absolute value function involved, as the function might behave differently on either side of a given point.