   Chapter 4.4, Problem 58E ### 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 +   ( tan 2 x ) x

To determine

To evaluate: The value of limx0+(tan2x)x .

Explanation

Let y=limx0+(tan2x)x . (1)

Take natural logarithm on both sides,

lny=ln(limx0+(tan2x)x)=limx0+(ln(tan2x)x)=limx0+(xln(tan2x))=limx0+ln(tan2x)1x

Therefore lny=limx0+ln(tan2x)1x . (2)

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

As x approaches 0+ , the numerator is,

ln(tan2x)=ln(tan(0))=ln(0)=

As x approaches 0+ , the denominator is,

1x=10=

Thus, limx0+ln(tan2x)1x= is in an indeterminate form.

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

limx0+ln(tan2x)1x=limx0+1tan2xsec2(2x)21x2=limx0+2x2sec2(2x)tan2x=limx0+2x2sec2(2x)sin2xcos2x=limx0+2x2sec(2x)sin2x

The above function can be separated as,

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