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, limx→∞logbxx1/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/n= limx→+∞1/( x⋅( in b ))x 1/n−1/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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