Math Problem Statement
Solution
The problem provided is about finding the largest such that if , then for the given function . The graph of the function near is shown, and we need to use this to determine the value of .
Step-by-Step Solution:
We are dealing with the function: and we are asked to find the largest such that:
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Rewriting the inequality: The condition is , which means:
Simplifying the expression: So we now need:
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Solving the inequality: This inequality can be written as: This implies that:
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Graphical Interpretation: We want to find the largest such that when is close to 2 (but not equal to 2), the function's values satisfy .
The zoomed-in part of the graph near shows a range of values where is between 0.99 and 1.01. From this visual approximation, it seems that the interval around for which this condition holds is small. From the graph, we can estimate that , since this is the range where the function is between 0.99 and 1.01.
Thus, the largest is approximately 0.005 based on the graph.
Do you have any questions or need further details?
Here are 5 related questions to deepen understanding:
- How can you calculate the exact value of algebraically instead of graphically?
- What does imply about the behavior of the function near ?
- How does the function behave as approaches 1 from the right and left?
- What are the implications of finding in the context of limits and continuity?
- How would the answer change if the inequality was ?
Tip: Always double-check your results graphically and algebraically to ensure consistency.
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Math Problem Analysis
Mathematical Concepts
Limits
Continuity
Inequalities
Formulas
f(x) = 1 / (x - 1)
|f(x) - 1| < 0.01
|2 - x| / |x - 1| < 0.01
Theorems
Limit definition of continuity
Epsilon-delta definition
Suitable Grade Level
Undergraduate Calculus