We are tasked with analyzing the graph of a function $h(x)$ given the following properties:

• $h(-1) = 4$
• $\lim_{x \to -1^-} h(x) = 2$
• $\lim_{x \to -1^+} h(x) = 3$
• $h(1) = \lim_{x \to 1^-} h(x) = 1$
• $\lim_{x \to 1^+} h(x) = 0$

Let's break down the conditions:

1. At $x = -1$:

   • $h(-1) = 4$: The graph passes through the point $(-1, 4)$.
   • $\lim_{x \to -1^-} h(x) = 2$: As $x$ approaches $-1$ from the left, the graph approaches the value $2$.
   • $\lim_{x \to -1^+} h(x) = 3$: As $x$ approaches $-1$ from the right, the graph approaches the value $3$.

2. At $x = 1$:

   • $h(1) = 1$: The graph passes through the point $(1, 1)$.
   • $\lim_{x \to 1^-} h(x) = 1$: As $x$ approaches $1$ from the left, the graph approaches the value $1$.
   • $\lim_{x \to 1^+} h(x) = 0$: As $x$ approaches $1$ from the right, the graph approaches the value $0$.

Matching these conditions to the graphs:

• Option A:

  • $h(-1) = 4$, correct.
  • $\lim_{x \to -1^-} h(x) = 2$ and $\lim_{x \to -1^+} h(x) = 3$, correct.
  • $h(1) = 1$ and $\lim_{x \to 1^-} h(x) = 1$, correct.
  • $\lim_{x \to 1^+} h(x) = 0$, correct.

• Option B:

  • At $x = -1$, the graph doesn't satisfy $h(-1) = 4$, so this is incorrect.

• Option C:

  • At $x = -1$, the graph does not satisfy $\lim_{x \to -1^-} h(x) = 2$, so this is incorrect.

• Option D:

  • At $x = 1$, the graph does not satisfy $\lim_{x \to 1^+} h(x) = 0$, so this is incorrect.

Conclusion:

Option A is the correct graph that satisfies all the given conditions.

Would you like more details on how we verified this? Here are 5 related questions:

1. What is the significance of left-hand and right-hand limits in graphing?
2. How does the graph behave at points of discontinuity?
3. Can a graph have different limits from the left and right at the same point?
4. Why is it important to check both the function's value and limits?
5. How do you sketch a graph given discontinuities?

Tip: When checking limits, always consider both the left-hand and right-hand behavior, especially around points where discontinuities or jumps occur.