Let's analyze the graph and determine the requested limits step by step. Here's how we'll approach the problem:

1. Problem Breakdown

a. $\lim_{x \to 0^-} h(x)$:

• Look at the behavior of the function as $x$ approaches $0$ from the left. The graph shows that as $x \to 0^-$, $h(x)$ approaches $4$.

b. $\lim_{x \to 0^+} h(x)$:

• Look at the behavior of the function as $x$ approaches $0$ from the right. The graph shows that as $x \to 0^+$, $h(x)$ approaches $2$.

c. $\lim_{x \to 0} h(x)$:

• This overall limit exists only if the left-hand limit ($4$) and right-hand limit ($2$) agree. Since they are different, the limit does not exist ($\text{DNE}$).

d. $h(-2)$:

• At $x = -2$, the graph shows that $h(-2) = -2$.

e. $\lim_{x \to 4} h(x)$:

• As $x$ approaches $4$, observe that the graph has a hole at $x = 4$, but $h(x)$ approaches $0$ from both sides. Thus, $\lim_{x \to 4} h(x) = 0$.

2. Answers

• (a) $\lim_{x \to 0^-} h(x) = 4$
• (b) $\lim_{x \to 0^+} h(x) = 2$
• (c) $\lim_{x \to 0} h(x) = \text{DNE}$
• (d) $h(-2) = -2$
• (e) $\lim_{x \to 4} h(x) = 0$

Would you like me to elaborate further on any part?

Here are 5 related questions for deeper understanding:

1. What is the difference between left-hand and right-hand limits, and why do they matter for continuity?
2. Why does the overall limit not exist when the one-sided limits differ?
3. How would you determine the continuity of $h(x)$ at $x = 0$?
4. Why is $\lim_{x \to 4} h(x)$ unaffected by the hole at $x = 4$?
5. What would happen if the function had no break at $x = 0$?

Tip: Always check both one-sided limits to confirm if a two-sided limit exists.