generate questions on Getting Limiting Values ​
from Graphs​ and show a explained answer key at the end

Here are five questions related to finding limiting values from graphs, followed by an explained answer key:

### Questions:

1. **Question 1:**
   Consider the graph of a function \( f(x) \). As \( x \to 2 \), the graph approaches a horizontal line at \( y = 3 \), but the function is undefined at \( x = 2 \). What is the limiting value of \( f(x) \) as \( x \) approaches 2?

2. **Question 2:**
   For the graph of \( g(x) \), it is observed that as \( x \to -\infty \), the graph gets closer and closer to the horizontal line \( y = -5 \). What is the limiting value of \( g(x) \) as \( x \to -\infty \)?

3. **Question 3:**
   The graph of a function \( h(x) \) shows that as \( x \to 4 \), the graph approaches \( y = 7 \) from both sides, but at \( x = 4 \), the value of the function is \( h(4) = 2 \). What is the limit of \( h(x) \) as \( x \to 4 \)?

4. **Question 4:**
   Looking at the graph of a function \( k(x) \), the function increases without bound as \( x \to 3^+ \) (from the right). What is the limit of \( k(x) \) as \( x \to 3^+ \)?

5. **Question 5:**
   From the graph of \( p(x) \), we see that as \( x \to -1 \) from the left, the function approaches 0, but as \( x \to -1 \) from the right, the function approaches 2. What can you say about the limit of \( p(x) \) as \( x \to -1 \)?

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### Explained Answer Key:

1. **Answer 1:**
   The limit of \( f(x) \) as \( x \to 2 \) is the value that the function approaches as \( x \) gets closer and closer to 2, regardless of whether the function is defined at \( x = 2 \) or not. From the graph, we observe that as \( x \to 2 \), the function approaches \( y = 3 \). Therefore, 
   \[
   \lim_{x \to 2} f(x) = 3.
   \]

2. **Answer 2:**
   The limit as \( x \to -\infty \) refers to the behavior of the function as \( x \) decreases without bound. The graph indicates that the function approaches the horizontal line \( y = -5 \). Therefore, 
   \[
   \lim_{x \to -\infty} g(x) = -5.
   \]

3. **Answer 3:**
   The limit of \( h(x) \) as \( x \to 4 \) depends on the values the function approaches as \( x \) gets closer to 4 from both sides. Although the function is defined at \( h(4) = 2 \), this value does not affect the limiting behavior. The graph shows that as \( x \) approaches 4, the function approaches \( y = 7 \). Thus, 
   \[
   \lim_{x \to 4} h(x) = 7.
   \]

4. **Answer 4:**
   The behavior of the function as \( x \to 3^+ \) (from the right) shows that the function increases without bound, which means that the function tends towards positive infinity. Therefore, 
   \[
   \lim_{x \to 3^+} k(x) = +\infty.
   \]

5. **Answer 5:**
   When the left-hand and right-hand limits of a function as \( x \to -1 \) are not equal, the overall limit at that point does not exist. In this case, the left-hand limit is \( 0 \) and the right-hand limit is \( 2 \). Since these limits are not the same, we conclude that 
   \[
   \lim_{x \to -1} p(x) \text{ does not exist}.
   \]

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### Follow-up Questions:

1. How do you determine the limit of a function if the function is undefined at the point of interest?
2. What is the difference between the left-hand limit and right-hand limit of a function?
3. Can a function have a limit at a point where it is not continuous? Explain why or why not.
4. How does the behavior of a function as \( x \to \infty \) or \( x \to -\infty \) relate to its asymptotes?
5. What role do discontinuities play when finding limits from graphs?

**Tip**: When evaluating limits from graphs, focus on the behavior of the function as it approaches the point from either side, rather than the actual value of the function at that point.

Generate questions on Getting Limiting Values from Graphs and show an explained answer key at the end.

Explore how to find limiting values from graphs, focusing on graphical interpretation of limits, asymptotic behavior, and discontinuities. This detailed guide includes questions and an explained answer key, suitable for students in Grades 11-12.

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