   Chapter 4.4, Problem 66E ### 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   ( 2 − x ) tan ( π x / 2 )

To determine

To evaluate: The value of limx1(2x)tan(πx2) .

Explanation

Given:

Let y=limx1(2x)tan(πx2) (1)

Calculation:

Take natural logarithm on both sides,

lny=ln(limx1(2x)tan(πx2))=limx1(ln(2x)tan(πx2))=limx1(tan(πx2)ln(2x))=limx1ln(2x)cot(πx2)

Therefore, lny=limx1ln(2x)cot(πx2) (2)

Obtain the value of the function as x approaches 1 .

As x approaches 1 , the numerator is,

ln(2x)=ln(21)=ln1=0

As x approaches 0+ , the denominator is,

cot(πx2)=cot(π2)=0

Thus, limx1ln(2x)cot(πx2)=00 is in an indeterminate form.

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

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