   Chapter 4.4, Problem 51E ### 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

# Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. lim x → 1 ( x x − 1 − 1 ln x )

To determine

To evaluate: The value of limx1(x1x1lnx) .

Explanation

Express the given function as, limx1(x1x1lnx)=limx1(xlnx(1x)(1x)lnx) .

Obtain the value of the function as x approaches 1 .

As x approaches 1 , the numerator is,

xlnx(1x)=1ln(1)(11)=100=0

As x approaches 1 , the denominator is,

(1x)lnx=(11)ln1=00=0

Thus, limx1(x1x1lnx)=00 is in an indeterminate form.

Therefore, apply L’Hospital’s Rule and obtain the limit as shown below.

limx1(xx11lnx)=limx1(xlnx(x1)(x1)lnx)=limx1(x1x+lnx1lnx+(x1)1x<

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