Let's analyze the graph and solve the limits and function values step by step. Based on the image you uploaded, I will interpret the graph and the questions as follows:

Problem breakdown:

• (a) $\lim_{x \to 2^-} f(x)$: The left-hand limit as $x$ approaches 2 from the left.
• (b) $\lim_{x \to 2^+} f(x)$: The right-hand limit as $x$ approaches 2 from the right.
• (c) $\lim_{x \to 2} f(x)$: The two-sided limit as $x$ approaches 2.
• (d) $f(2)$: The value of the function at $x = 2$.
• (e) $\lim_{x \to 4} f(x)$: The limit as $x$ approaches 4.

Analysis of the graph:

For $x = 2$ (Parts a, b, c, d):

• From the graph, as $x \to 2^-$, the function $f(x)$ approaches $0$.
• As $x \to 2^+$, the function $f(x)$ approaches $2$.
• The two-sided limit $\lim_{x \to 2} f(x)$ does not exist (DNE) because the left-hand and right-hand limits are not equal.
• $f(2)$ appears to be $f(2) = 0$, as the function touches the x-axis at $x = 2$.

For $x = 4$ (Part e):

• As $x \to 4$, the function $f(x)$ approaches $2$. The graph is continuous around $x = 4$, so the limit exists and is $2$.

Final Answers:

• (a) $\lim_{x \to 2^-} f(x) = 0$
• (b) $\lim_{x \to 2^+} f(x) = 2$
• (c) $\lim_{x \to 2} f(x) = \text{DNE}$ (since the left and right limits are not equal)
• (d) $f(2) = 0$
• (e) $\lim_{x \to 4} f(x) = 2$

Would you like further clarification or more details on any part of this?

Here are 5 related questions that can expand on the concepts discussed:

1. What conditions must be met for the two-sided limit $\lim_{x \to c} f(x)$ to exist?
2. How do you find the limit of a function when given its graph?
3. What is the difference between a removable discontinuity and a jump discontinuity?
4. How can the value of a function at a point differ from its limit at that point?
5. Why does the limit not exist at $x = 2$ in this case?

Tip: When dealing with piecewise functions or discontinuities, always check left-hand and right-hand limits separately to confirm the overall limit.