   Chapter 2.6, Problem 74E

Chapter
Section
Textbook Problem

For the limit lim x → ∞ x ln x = ∞ illustrate Definition 9 by finding a value of N that corresponds to M = 100.Definition 9 To determine

To find: The value of N for the limit limxxlnx= that correspond to M=100.

Explanation

Definition: Let f be a function defined on some interval (a,). Then limxf(x)= means that for every positive number M there is a corresponding positive number N such that if x>N then f(x)>M.

Proof:

Let f(x)=xlnx and M=100.

By the definition,

f(x)>Mxlnx>100

Squaring on both sides.

xlnx>(100)2xlnx>10,000xlnx10,000>1xlnx>10,000

Differentiate on both sides.

[x×1x+(1)×lnx]>0[1+lnx]>0

Take exponential on both sides

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