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Show that f(r) = O(|x – xo|*) and g(x) = o(]x – roP) imply f. g(x) = o(|x – xo]k+i). - -

Show that f(r) = O(|x – xo|*) and g(x) = o(]x – roP) imply f. g(x) = o(|x – xo]k+i). - -

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 Statement for Educational Purposes** **Objective:** Demonstrate the following mathematical relationship: Given that \( f(x) = O(|x - x_0|^k) \) and \( g(x) = o(|x - x_0|^j) \), prove that \( f \cdot g(x) = o(|x - x_0|^{k+j}) \). *Explanation of Terms:* - \( O \) notation (Big O) is used to describe an upper bound on the growth rate of a function as \( x \) approaches \( x_0 \). - \( o \) notation (little o) describes a function that grows significantly slower than another as \( x \) approaches \( x_0 \). **Instructions for Proof:** 1. **Interpret the assertion:** - \( f(x) = O(|x - x_0|^k) \) implies that there exists a constant \( C > 0 \) and a neighborhood around \( x_0 \) such that \( |f(x)| \leq C|x - x_0|^k \) for values of \( x \) in this neighborhood. - \( g(x) = o(|x - x_0|^j) \) implies that for any \( \epsilon > 0 \), there exists a neighborhood around \( x_0 \) such that \( |g(x)| < \epsilon |x - x_0|^j \) for values of \( x \) in this neighborhood. 2. **Combine the functions:** - Analyze the product \( f(x) \cdot g(x) \). - Utilize the definitions of \( O \) and \( o \) to prove that this product satisfies the condition \( o(|x - x_0|^{k+j}) \). **Note:** This type of proof typically involves manipulations and inequalities to show that the product of \( f(x) \) and \( g(x) \) indeed behaves as described by the little o notation for the combined powers \( k+j \).
Transcribed Image Text:**Problem Statement for Educational Purposes** **Objective:** Demonstrate the following mathematical relationship: Given that \( f(x) = O(|x - x_0|^k) \) and \( g(x) = o(|x - x_0|^j) \), prove that \( f \cdot g(x) = o(|x - x_0|^{k+j}) \). *Explanation of Terms:* - \( O \) notation (Big O) is used to describe an upper bound on the growth rate of a function as \( x \) approaches \( x_0 \). - \( o \) notation (little o) describes a function that grows significantly slower than another as \( x \) approaches \( x_0 \). **Instructions for Proof:** 1. **Interpret the assertion:** - \( f(x) = O(|x - x_0|^k) \) implies that there exists a constant \( C > 0 \) and a neighborhood around \( x_0 \) such that \( |f(x)| \leq C|x - x_0|^k \) for values of \( x \) in this neighborhood. - \( g(x) = o(|x - x_0|^j) \) implies that for any \( \epsilon > 0 \), there exists a neighborhood around \( x_0 \) such that \( |g(x)| < \epsilon |x - x_0|^j \) for values of \( x \) in this neighborhood. 2. **Combine the functions:** - Analyze the product \( f(x) \cdot g(x) \). - Utilize the definitions of \( O \) and \( o \) to prove that this product satisfies the condition \( o(|x - x_0|^{k+j}) \). **Note:** This type of proof typically involves manipulations and inequalities to show that the product of \( f(x) \) and \( g(x) \) indeed behaves as described by the little o notation for the combined powers \( k+j \).
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