   Chapter 4.4, Problem 42E ### 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 → a + cos x ln ( x − a ) ln ( e x − e a )

To determine

To evaluate: The value of limxa+cosxln(xa)ln(exea) .

Explanation

Consider given limit function as product of two functions as below,

limxa+cosxln(xa)ln(exea)=limxa+cosxlimxa+ln(xa)ln(exea) (1)

Consider the limit function limxa+ln(xa)ln(exea) .

Obtain the value of the function as x approaches a+ .

As x approaches a+ , the numerator is,

ln(xa)=ln(aa)=ln(0)=

And the value of denominator is,

ln(exea)=ln(eaea)=ln(0)=

Thus, limxa+ln(xa)ln(exea)= is in an indeterminate form.

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

limx1ln(xa)ln(exea)=limxa+1xaexexea=limxa+1xaexeaex=limxa+1exexeaxa=limxa+1exlimxa+exeaxa

Therefore limxa+ln(xa)ln(exea)=lim1exlimxa+exeaxa (2)

Consider limxa+exeaxa

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