6. Guess the smallest n such that the function x4x³ + 5x + 3 log x 2x² + x + 1 is O(x") and then use the definition of O to prove your conjecture.

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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**6. Guess the smallest \( n \) such that the function**

\[
\frac{x^4 - x^3 + 5x + 3 \log x}{2x^2 + x + 1}
\]

**is \( O(x^n) \) and then use the definition of \( O \) to prove your conjecture.**

**Explanation:**

The problem requires determining the smallest power \( n \) such that the given function is bounded by \( O(x^n) \) in the context of asymptotic notation, which describes the behavior of functions as they approach a particular limit or infinity. The numerator of the function is dominated by the largest term \( x^4 \), while the denominator is dominated by \( 2x^2 \). The objective is to assess the dominant term in the expression and confirm the asymptotic behavior analytically.
Transcribed Image Text:**6. Guess the smallest \( n \) such that the function** \[ \frac{x^4 - x^3 + 5x + 3 \log x}{2x^2 + x + 1} \] **is \( O(x^n) \) and then use the definition of \( O \) to prove your conjecture.** **Explanation:** The problem requires determining the smallest power \( n \) such that the given function is bounded by \( O(x^n) \) in the context of asymptotic notation, which describes the behavior of functions as they approach a particular limit or infinity. The numerator of the function is dominated by the largest term \( x^4 \), while the denominator is dominated by \( 2x^2 \). The objective is to assess the dominant term in the expression and confirm the asymptotic behavior analytically.
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