From the image you provided, I can observe the graph of the function $f(x)$, and there are four limit questions asked based on the graph. Let's solve them by analyzing the behavior of the graph at the given points:

(a) $\lim_{x \to 4} f(x)$

• From the graph, as $x \to 4$, the function approaches the value of $y = 5$. There is no discontinuity or jump at this point. $\boxed{5}$

(b) $\lim_{x \to 2^-} f(x)$

• For $x \to 2$ from the left ($2^-$), the graph appears to approach $y = 0$. So, the left-hand limit is: $\boxed{0}$

(c) $\lim_{x \to 2^+} f(x)$

• For $x \to 2$ from the right ($2^+$), the graph shows a jump, and the function value jumps to $y = 5$. So, the right-hand limit is: $\boxed{5}$

(d) $\lim_{x \to 2} f(x)$

• Since the left-hand limit and the right-hand limit as $x \to 2$ are not equal (left-hand limit is 0 and right-hand limit is 5), the two-sided limit does not exist (DNE). $\boxed{\text{DNE}}$

Summary of the Limits:

• $\lim_{x \to 4} f(x) = 5$
• $\lim_{x \to 2^-} f(x) = 0$
• $\lim_{x \to 2^+} f(x) = 5$
• $\lim_{x \to 2} f(x) = \text{DNE}$

Would you like further details or explanations on any part? Here are 5 related questions to explore:

1. What is the definition of a one-sided limit?
2. How does continuity at a point relate to the existence of a two-sided limit?
3. Can a function have different one-sided limits and still be continuous at that point?
4. What is the significance of a jump discontinuity for limits?
5. How do we formally prove that a limit does not exist?

Tip: Always check both one-sided limits before concluding whether a two-sided limit exists!