Math Problem Statement
Suppose that f is integrable on [0, x] for every x > 0, and that limx→∞ f (x) = a. Prove that lim x→∞ 1 x ∫ x 0 f (t) dt = a . Hint: For N > 0 (to be chosen later) and x > N , write x = N + y and split the integral as ∫ N +y 0 f (t) dt = ∫ N 0 f (t) dx + ∫ N +y N f (t) dt . Take N large and compare the second integral with a · y
Solution
To prove that
given that is integrable on for every and that , we can follow the hint provided and proceed step-by-step.
Step 1: Split the integral
For , write , where , and then split the integral as suggested:
Thus, we have
Step 2: Deal with the first integral
The first integral is independent of and thus becomes negligible as . Specifically,
Step 3: Estimate the second integral
Now, consider the second integral:
For large , since , for sufficiently large , is close to for all . More formally, for any , there exists some such that for all , .
Thus, for , we can estimate:
Since for , we have
Therefore,
Step 4: Divide by
Now consider the full expression:
From the previous steps, we have:
- as ,
- .
Since and as , , we conclude that
Step 5: Conclusion
Therefore,
Hence, we have shown that:
This completes the proof.
Would you like more details on any part of the proof, or do you have additional questions?
Here are 5 related questions that might help further your understanding:
- How do we use the definition of the limit to prove that ?
- What is the significance of the integral test for convergence in this context?
- How does the concept of Riemann integrability relate to the problem at hand?
- What would happen if did not tend to as ?
- How does the concept of a "local error" in the integral help us bound the difference between the two integrals?
Tip: The key idea in these kinds of proofs is breaking the problem into manageable parts and handling the asymptotic behavior of the function carefully.
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Math Problem Analysis
Mathematical Concepts
Integral Calculus
Limits
Asymptotic Behavior
Riemann Integrability
Formulas
lim x→∞ 1/x ∫ x 0 f(t) dt = a
Theorems
Fundamental Theorem of Calculus
Squeeze Theorem (for limits)
Suitable Grade Level
Grades 12-14 (College/University level)
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