   Chapter 11.4, Problem 49ES ### 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.49. a. Let b be any real number greater than 1. Use L’Hôpital’s rule and mathematical induction to prove that for every integer n ≥ 1 , lim x → ∞ x a b x = 0. b. Use the result of part (a) and the definitions of limit and of O-notation to prove that x n is O ( b x ) for any integer n ≥ 1 .

To determine

(a)

To prove:

Use L’Hôpital’srule and mathematical induction to prove that for allintegers n = 1,

limxxnbx=0.

Explanation

Given information:

Let b be any real number greater than 1.

Proof:

Given: b is a real number greater than 1

PROOF BY INDUCTION:

Let P(n) be "limx+xnbx=0"

Basis step: n = 1

limx+x1bx=limx+xbx            limx+x=+ and limx+bx=+ as b>1

=limx+1bx( In b)                                           I'Hopital's rule                                      ddx(x)=1 and ddx(bx)=bx(In b)

=1In blimx+1bx

=1In b0                                                          limx+bx=+

= 0

Thus P (1) is true.

Inductive step:

Let P ( k ) be true, thus limx+xkbx=0 where k is an integer such that k1

To determine

(b)

Use the result of part (a) and the definitions of limit andof O -notation to prove that xn is O(bx) for any integern = 1.

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