   Chapter 4.4, Problem 65E ### 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 +   ( 4 x + 1 ) cot x

To determine

To evaluate: The value of limx0+(4x+1)cotx .

Explanation

Given:

Let y=limx0+(4x+1)cotx . (1)

Calculation:

Take natural logarithm on both sides,

lny=ln(limx0+(4x+1)cotx)=limx0+(ln(4x+1)cotx)=limx0+(cotxln(4x+1))=limx0+(cotxln(4x+1))=limx0+ln(4x+1)tanx

Therefore, lny=limx0+ln(4x+1)tanx . (2)

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

As x approaches 0+ , the numerator is,

ln(4x+1)=ln(40+1)=ln1=0

As x approaches 0+ , the denominator is,

tanx=tan0=0

Thus, limx0+ln(4x+1)tanx=00 is in an indeterminate form

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