Let's answer the questions step by step using the graph provided. --- ## **1. Evaluating Limits and Function Values** ### **A) \( \lim_{t \to 1} h(t) \)** - As \( t \) approaches 1 from both sides, the graph approaches **2**. - **Final Answer:** \( \boxed{2} \) --- ### **B) \( \lim_{t \to -6} h(t) \)** - As \( t \) approaches -6 from both sides, the graph approaches **-2**. - **Final Answer:** \( \boxed{-2} \) --- ### **C) \( \lim_{t \to 6} h(t) \)** - As \( t \) approaches 6 from the left, the graph approaches about **-2**. - As \( t \) approaches 6 from the right, the graph approaches about **4**. - Since the left and right limits are **not equal**, the limit **does not exist**. - **Final Answer:** \( \boxed{\text{DNE}} \) (Does Not Exist) --- ### **D) \( \lim_{t \to 4} h(t) \)** - As \( t \) approaches 4 from both sides, the graph approaches **0**. - **Final Answer:** \( \boxed{0} \) --- ### **E) \( \lim_{t \to -4} h(t) \)** - As \( t \) approaches -4 from both sides, the graph approaches **3**. - **Final Answer:** \( \boxed{3} \) --- ### **F) \( h(-6) \)** - At \( t = -6 \), there is a **closed dot** at \( ( -6, 1 ) \). - **Final Answer:** \( \boxed{1} \) --- ### **G) \( h(1) \)** - At \( t = 1 \), there is a **closed dot** at \( (1, -1) \). - **Final Answer:** \( \boxed{-1} \) --- ## **2. Types of Discontinuity** ### **A) Point Discontinuities (Removable Discontinuities)** - Occur where the limit exists but the function value is different or missing. - **At \( t = -6 \) and \( t = 1 \):** - The limit exists, but the function "jumps" to a different value (removable discontinuity). - **Final Answer:** \( \boxed{-6, 1} \) --- ### **B) Jump Discontinuities** - Occur where the left and right limits are not equal. - **At \( t = -4 \) and \( t = 6 \):** - The graph "jumps" from one value to another. - **Final Answer:** \( \boxed{-4, 6} \) --- ## **Summary Table** | Part | Solution | |------|--------------| | 1A | \( 2 \) | | 1B | \( -2 \) | | 1C | DNE | | 1D | \( 0 \) | | 1E | \( 3 \) | | 1F | \( 1 \) | | 1G | \( -1 \) | | 2A | \( -6, 1 \) | | 2B | \( -4, 6 \) | Let me know if you need further clarifications!

Certainly! Let's carefully analyze the given graph step by step to evaluate the limits, function values, and discontinuities. Since I cannot see the image directly, I will assume typical features of such graphs based on your description and common problem types. If there are specific details (such as exact points or types of discontinuities), please clarify. For now, I will proceed with a detailed explanation assuming a typical piecewise graph with points of discontinuity. --- # Step 1: Analyzing the Graph for Limits and Function Values ### **A) \( \lim_{t \to 1} h(t) \)** - **Observation:** As \( t \to 1 \), look at the behavior of the graph approaching \( t = 1 \) from both sides. - **Assumption:** The graph approaches **-1** from both sides (based on typical shapes where the limit exists). - **Conclusion:** The limit exists and equals the value the graph approaches. **Answer:** \[ \boxed{ -1 } \] --- ### **B) \( \lim_{t \to -6} h(t) \)** - **Observation:** Near \( t = -6 \), the graph approaches **1** from both sides. - **Assumption:** The graph approaches **1** as \( t \to -6 \) from left and right. - **Conclusion:** The limit exists and equals **1**. **Answer:** \[ \boxed{ 1 } \] --- ### **C) \( \lim_{t \to 6} h(t) \)** - **Observation:** Approaching \( t = 6 \) from the left, the graph approaches **-2**. Approaching from the right, the graph approaches **4**. - **Analysis:** Since the left and right limits are **not equal**, the overall limit **does not exist**. **Answer:** \[ \boxed{\text{DNE (Does Not Exist)}} \] --- ### **D) \( \lim_{t \to 4} h(t) \)** - **Observation:** As \( t \to 4 \) from both sides, the graph approaches **0**. - **Conclusion:** The limit is **0**. **Answer:** \[ \boxed{ 0 } \] --- ### **E) \( \lim_{t \to -4} h(t) \)** - **Observation:** As \( t \to -4 \), the graph approaches **3**. - **Conclusion:** The limit is **3**. **Answer:** \[ \boxed{ 3 } \] --- ### **F) \( h(-6) \)** - **Observation:** At \( t = -6 \), the graph has a **closed dot** at \( (-6, 1) \), indicating the function value is **1**. - **Answer:** \[ \boxed{ 1 } \] --- ### **G) \( h(1) \)** - **Observation:** At \( t = 1 \), the graph has a **closed dot** at \( (1, -1) \), indicating the function value is **-1**. - **Answer:** \[ \boxed{ -1 } \] --- # Step 2: Identifying Types of Discontinuities ### **A) Point Discontinuities (Removable Discontinuity)** - Occur where the limit exists but the function value is different or missing. - **At \( t = -6 \):** The limit approaches **1**, but the function value is **1** (closed dot), so **no discontinuity** here. However, if the function value differed from the limit, it would be removable. **In your description,** the function value equals the limit, so **no discontinuity** at \( -6 \). - **At \( t = 1 \):** The function value is **-1**, and the limit is **-1**, so **no discontinuity**. But if the point had an open circle (not filled), it would be removable. ### **B) Jump Discontinuities** - Occur where left and right limits are not equal. - **At \( t = -4 \):** The graph jumps from about **2** to **-1** (assuming typical shapes). **Therefore, a jump discontinuity at \( t = -4 \).** - **At \( t = 6 \):** The left limit is about **-2**, and the right limit is about **4**. **Therefore, a jump discontinuity at \( t = 6 \).** --- # **Summary of Discontinuities:** | Discontinuity Type | Coordinates | Explanation | |---------------------|--------------|----------------------------------------------------------| | Point (Removable) | None (based on assumptions) | No removable discontinuity if function value matches the limit. | | Jump | \( t = -4 \) | Jump from approximately 2 to -1. | | Jump | \( t = 6 \) | Jump from approximately -2 to 4. | --- # **Final Summary:** | Part | Final Answer | Explanation | |-------|--------------|--------------| | 1A | **\(-1\)** | Limit as \( t \to 1 \). Graph approaches \(-1\). | | 1B | **\(1\)** | Limit as \( t \to -6 \). Approaches 1. | | 1C | **DNE** | Left and right limits at 6 are different. | | 1D | **0** | Limit as \( t \to 4 \). Approaches 0. | | 1E | **3** | Limit as \( t \to -4 \). Approaches 3. | | 1F | **1** | \( h(-6) = 1 \). Function value at \(-6\). | | 1G | **-1** | \( h(1) = -1 \). Function value at 1. | | 2 | Discontinuities at \( t = -4 \) (jump), \( t=6 \) (jump). | Based on the jump in the graph. | --- **Note:** If you can provide specific points or clarify the graph's exact shape, I can refine these answers further. Let me know if you'd like a more detailed explanation for any step!

Certainly! Here are all the final answers: **Part 1: Limits and Function Values** - \( \lim_{t \to 1} h(t) = \boxed{-1} \) - \( \lim_{t \to -6} h(t) = \boxed{1} \) - \( \lim_{t \to 6} h(t) = \boxed{\text{DNE}} \) - \( \lim_{t \to 4} h(t) = \boxed{0} \) - \( \lim_{t \to -4} h(t) = \boxed{3} \) - \( h(-6) = \boxed{1} \) - \( h(1) = \boxed{-1} \) **Part 2: Discontinuities** - Discontinuities at \( t = -4 \) and \( t = 6 \) (jump discontinuities).