   Chapter 4.4, Problem 53E ### 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 → 0 + ( 1 x − 1 e x − 1 )

To determine

To evaluate: The value of limx0+(1x1ex1) .

Explanation

Express the given function as, limx0+(1x1ex1)=limx0+((ex1)xx(ex1)) .

Obtain the value of the function as x approaches 0+ .

As x approaches 0+ , the numerator is,

ex1x=e010=110=0

As x approaches 0+ , the denominator is,

x(ex1)=0(e01)=0(11)=0

Thus, limx1(1x1ex1)=00 is in an indeterminate form.

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

limx0+(1x1ex1)=limx0+((ex1)xx(ex1))=limx0+(ex1xex+(ex1))

Divide each term by ex ,

limx0+(ex1xex+(e

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