   Chapter 11.4, Problem 50ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Exercises 49 and 50 use L’Hôpital’s rule from calculus.50. a. Let b be any real number greater than 1. Use L’Hôpital’s rule to prove that for every integer n ≥ 1 , lim x → ∞ log b x x 1 / n = 0. b. Use the result of part (a) and the definitions of limit and of O-notation to prove that log b x is O ( x 1 / n ) for any integer n ≥ 1 .

To determine

(a)

To prove: Use L'Hopital's  rule to prove that for all integers n = 1, limxlogbxx1/n=0.

Explanation

Given information:

Let b be any real number greater than 1.

Proof:

Let n be a positive integer and let b be a real number greater than 1.

limx+x=+ and limx+x1/n=+

limx+ logbxx 1/nlimx+1/( x( in b ))x 1/n1/n                                          I'Hopital's rule                                      ddx( logbx)=1x( In b)

To determine

(b)

Use the result of part (a) and the definitions of limit andof O -notation to prove that logbx is

O(x1/n) for anyinteger n = 1.

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