Math Problem Statement
generate questions on Getting Limiting Values from Graphs and show a explained answer key at the end
Solution
Here are five questions related to finding limiting values from graphs, followed by an explained answer key:
Questions:
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Question 1: Consider the graph of a function . As , the graph approaches a horizontal line at , but the function is undefined at . What is the limiting value of as approaches 2?
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Question 2: For the graph of , it is observed that as , the graph gets closer and closer to the horizontal line . What is the limiting value of as ?
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Question 3: The graph of a function shows that as , the graph approaches from both sides, but at , the value of the function is . What is the limit of as ?
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Question 4: Looking at the graph of a function , the function increases without bound as (from the right). What is the limit of as ?
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Question 5: From the graph of , we see that as from the left, the function approaches 0, but as from the right, the function approaches 2. What can you say about the limit of as ?
Explained Answer Key:
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Answer 1: The limit of as is the value that the function approaches as gets closer and closer to 2, regardless of whether the function is defined at or not. From the graph, we observe that as , the function approaches . Therefore,
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Answer 2: The limit as refers to the behavior of the function as decreases without bound. The graph indicates that the function approaches the horizontal line . Therefore,
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Answer 3: The limit of as depends on the values the function approaches as gets closer to 4 from both sides. Although the function is defined at , this value does not affect the limiting behavior. The graph shows that as approaches 4, the function approaches . Thus,
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Answer 4: The behavior of the function as (from the right) shows that the function increases without bound, which means that the function tends towards positive infinity. Therefore,
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Answer 5: When the left-hand and right-hand limits of a function as are not equal, the overall limit at that point does not exist. In this case, the left-hand limit is and the right-hand limit is . Since these limits are not the same, we conclude that
Follow-up Questions:
- How do you determine the limit of a function if the function is undefined at the point of interest?
- What is the difference between the left-hand limit and right-hand limit of a function?
- Can a function have a limit at a point where it is not continuous? Explain why or why not.
- How does the behavior of a function as or relate to its asymptotes?
- 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.
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Math Problem Analysis
Mathematical Concepts
Limits
Graphical Interpretation of Limits
Asymptotic Behavior
Discontinuities
Formulas
lim_{x \to c} f(x)
Left-hand limit: lim_{x \to c^-} f(x)
Right-hand limit: lim_{x \to c^+} f(x)
Theorems
Limit Laws
Continuity and Discontinuity Theorems
Squeeze Theorem
Suitable Grade Level
Grades 11-12