Math Problem Statement
Solution
The problem in the image asks to formally prove that:
Given that is arbitrary, and with a choice of , the task is to determine the smallest value of that would satisfy the proof.
Step-by-Step Approach:
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Limit Setup: To prove , we need to show that for every , there exists a such that if , then .
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Simplify the Expression: We start by simplifying : Factor the expression: Therefore:
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Establish a Bound for : Since , we know . Thus, lies between 5 and 7: Therefore, lies between 12 and 14, which gives us the bound:
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Choosing : Now we need to ensure that . Using the bound on , we have: To satisfy , it is enough to have:
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Conclusion: Thus, we choose . The smallest value of that satisfies the proof is 14.
Final Result:
The smallest value of that would satisfy the proof is .
Would you like more details or have any questions?
Here are five related questions to deepen your understanding:
- How would the limit change if the function were instead of ?
- Can you explain why we need to bound in the proof?
- How would the choice of change if were larger than 14?
- What happens to the limit if instead of ?
- Why do we consider the minimum of and 1 when choosing ?
Tip: In epsilon-delta proofs, always simplify and factor the function as much as possible to clearly see how relates to .
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Math Problem Analysis
Mathematical Concepts
Limits
Epsilon-Delta Proof
Algebra
Formulas
lim_{x → c} f(x) = L (Epsilon-Delta Definition of a Limit)
(x^2 + x) - 42 = (x - 6)(x + 7)
Theorems
Epsilon-Delta Definition of a Limit
Suitable Grade Level
College level (Calculus I or Introductory Analysis)
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