   Chapter 4.4, Problem 86E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Suppose f is a positive function. If lim x → a f ( x ) = 0 and lim x → a g ( x ) = ∞ , show that lim x → a   [ f ( x ) ] g ( x ) = 0 This shows that 0∞ is not an indeterminate form.

To determine

To show: For a positive function f, if limxaf(x)=0 and limxag(x)= then limxa[f(x)]g(x)=0 .

Explanation

Proof:

Given:

The function f is a positive function and limxaf(x)=0 , limxag(x)= .

Let, y=[f(x)]g(x) .

Then, lny=g(x)lnf(x) .

Since f is a positive function, lnf(x) is well defined. So,

limxalny=limxag(x)lnf(x)=limxag(x)ln(limxaf(x))=ln(0)=

Simplify further as follows

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