   Chapter 4.4, Problem 41E ### 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 cos x − 1 + 1 2 x 2 x 4

To determine

To evaluate: The value of limx0cosx1+12x2x4 .

Explanation

Obtain the value of the function as x approaches 0 .

As x approaches 0 , the numerator value becomes,

cosx1+12x2=cos01+12(0)2=11+0=0

And the denominator value becomes,

x4=04=0

Thus, limx0cosx1+12x2x4=00 is in an indeterminate form.

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

limx0cosx1+12x2x4=limx0sinx+x4x3

Again as x approaches 0 , the numerator is,

sinx+x=sin0+0=0

And as x approaches 0 , the denominator is,

4x3=4(0)3=0

Thus, limx0sinx+x4x3=00 is again in an indeterminate form.

Apply L’Hospital’s Rule again and obtain the limit

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