https://file.aiquickdraw.com/mathbot/file/1730226804357efj7q4n4.jpg

Let's analyze the graph to answer each of the given limits and function values.

### Given Information

We need to estimate values for:
1. \( \lim_{{x \to -2^-}} f(x) \)
2. \( \lim_{{x \to -2^+}} f(x) \)
3. \( \lim_{{x \to -2}} f(x) \)
4. \( f(-2) \)
5. \( \lim_{{x \to \infty}} f(x) \)

Based on the graph, let's proceed with each part.

### Solutions

**(a) \( \lim_{{x \to -2^-}} f(x) \):**
   - As \( x \) approaches \(-2\) from the left side (\( x \to -2^- \)), the value of \( f(x) \) approaches approximately \(1\).

   **Answer**: \( \lim_{{x \to -2^-}} f(x) = 1 \)

**(b) \( \lim_{{x \to -2^+}} f(x) \):**
   - As \( x \) approaches \(-2\) from the right side (\( x \to -2^+ \)), \( f(x) \) also appears to approach \(1\).

   **Answer**: \( \lim_{{x \to -2^+}} f(x) = 1 \)

**(c) \( \lim_{{x \to -2}} f(x) \):**
   - Since both the left-hand limit and the right-hand limit as \( x \to -2 \) approach \(1\), the two-sided limit also equals \(1\).

   **Answer**: \( \lim_{{x \to -2}} f(x) = 1 \)

**(d) \( f(-2) \):**
   - From the graph, it appears there is a filled dot at \((-2, 1)\), indicating \( f(-2) = 1\).

   **Answer**: \( f(-2) = 1 \)

**(e) \( \lim_{{x \to \infty}} f(x) \):**
   - As \( x \) approaches infinity (\( x \to \infty \)), \( f(x) \) seems to be decreasing towards \(0\).

   **Answer**: \( \lim_{{x \to \infty}} f(x) = 0 \)

### Summary of Answers
1. \( \lim_{{x \to -2^-}} f(x) = 1 \)
2. \( \lim_{{x \to -2^+}} f(x) = 1 \)
3. \( \lim_{{x \to -2}} f(x) = 1 \)
4. \( f(-2) = 1 \)
5. \( \lim_{{x \to \infty}} f(x) = 0 \)

Would you like a more detailed explanation of any part?

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Here are some related questions to expand your understanding:
1. What is the significance of one-sided limits in determining two-sided limits?
2. How does the existence of a point at \(x = -2\) affect the value of the limit?
3. What happens to the limit if the function approaches different values from the left and right?
4. How would you interpret a limit as \(x \to -\infty\) for the function \(f(x)\)?
5. Why is it important to observe both the left-hand and right-hand limits for continuity?

**Tip:** When estimating limits from a graph, always check if the function approaches the same value from both sides for a two-sided limit!

Use the graph of the function g shown to estimate the indicated limits and function values.

Learn how to estimate one-sided and two-sided limits and function values from a graph. This problem explores key calculus concepts, including limits, continuity, and graph analysis. Suitable for high school calculus students.

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