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2.3 Find the least positive integer n such that x¹/(x² + 1) is O(x^), and then show that it is O(x") for that n

2.3 Find the least positive integer n such that x¹/(x² + 1) is O(x^), and then show that it is O(x") for that n

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem 2.3:** Find the least positive integer \( n \) such that \( \frac{x^4}{x^2 + 1} \) is \( O(x^n) \), and then show that it is \( O(x^n) \) for that \( n \). --- ### Explanation: This problem involves asymptotic analysis using Big O notation to determine the growth rate of the function \( \frac{x^4}{x^2 + 1} \) compared to a power of \( x \). Your task is to: 1. Identify the smallest integer \( n \) such that the given function can be bounded above by a constant multiple of \( x^n \) for sufficiently large values of \( x \). 2. Demonstrate mathematically that this function indeed conforms to \( O(x^n) \).
Transcribed Image Text:**Problem 2.3:** Find the least positive integer \( n \) such that \( \frac{x^4}{x^2 + 1} \) is \( O(x^n) \), and then show that it is \( O(x^n) \) for that \( n \). --- ### Explanation: This problem involves asymptotic analysis using Big O notation to determine the growth rate of the function \( \frac{x^4}{x^2 + 1} \) compared to a power of \( x \). Your task is to: 1. Identify the smallest integer \( n \) such that the given function can be bounded above by a constant multiple of \( x^n \) for sufficiently large values of \( x \). 2. Demonstrate mathematically that this function indeed conforms to \( O(x^n) \).
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