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Show that f(x) = 0(\x – #o|*) and g(x) f • g(x) = o(\x – xol*+i). o(lx – xo|2) imply -

Show that f(x) = 0(\x – #o|*) and g(x) f • g(x) = o(\x – xol*+i). o(lx – xo|2) imply -

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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**Mathematical Problem Description:** Show that \( f(x) = O(|x - x_0|^k) \) and \( g(x) = o(|x - x_0|^j) \) imply \( f \cdot g(x) = o(|x - x_0|^{k+j}) \). **Explanation:** In this problem, we are asked to demonstrate a relationship involving the asymptotic notation Big O and little o. Specifically, it states: - \( f(x) = O(|x - x_0|^k) \): This means that the function \( f(x) \) is bounded by a constant multiple of \(|x - x_0|^k\) for values of \(x\) near \(x_0\). - \( g(x) = o(|x - x_0|^j) \): This indicates that the function \( g(x) \) grows much slower than \(|x - x_0|^j\) as \(x\) approaches \(x_0\). The goal is to prove that the product \( f \cdot g(x) \) behaves asymptotically like \( o(|x - x_0|^{k+j}) \). This means that the growth rate of the product is negligible compared to \(|x - x_0|^{k+j}\) when \(x\) is near \(x_0\).
Transcribed Image Text:**Mathematical Problem Description:** Show that \( f(x) = O(|x - x_0|^k) \) and \( g(x) = o(|x - x_0|^j) \) imply \( f \cdot g(x) = o(|x - x_0|^{k+j}) \). **Explanation:** In this problem, we are asked to demonstrate a relationship involving the asymptotic notation Big O and little o. Specifically, it states: - \( f(x) = O(|x - x_0|^k) \): This means that the function \( f(x) \) is bounded by a constant multiple of \(|x - x_0|^k\) for values of \(x\) near \(x_0\). - \( g(x) = o(|x - x_0|^j) \): This indicates that the function \( g(x) \) grows much slower than \(|x - x_0|^j\) as \(x\) approaches \(x_0\). The goal is to prove that the product \( f \cdot g(x) \) behaves asymptotically like \( o(|x - x_0|^{k+j}) \). This means that the growth rate of the product is negligible compared to \(|x - x_0|^{k+j}\) when \(x\) is near \(x_0\).
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