   Chapter 4.4, Problem 30E ### 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 x − sin x x − tan x

To determine

To evaluate: The value of limx0xsinxxtanx .

Explanation

Obtain the value of the function as x approaches 0 .

As x approaches 0 , the numerator is,

xsinx=0sin0=00=0

As x approaches 0 , the denominator is,

xtanx=0tan0=00=0

Thus, limx0xsinxxtanx=00 is in an indeterminate form.

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

limx0xsinxxtanx=limx01cosx1sec2x

As x approaches 0 , the numerator is,

1cosx=1cos0=11=0

As x approaches 0 , the denominator is,

1sec2x=1sec20=11=0

Thus, limx01cosx1sec2x=00 is again in an indeterminate form

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