   Chapter 4.4, Problem 34E ### 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 m x − cos n x x 2

To determine

To evaluate: The value of limx0cosmxcosnxx2 .

Explanation

Obtain the value of the function as x approaches 0 .

As x approaches 0 , the numerator is ,

cosmxcosnx=cosm0cosn0=cos0cos0=11=0

And the denominator x2 approaches to 0 .

Thus, limx0cosmxcosnxx2=00 is in an indeterminate form.

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

limx0cosmxcosnxx2=limx0msinmx+nsinnx2x

Again as x approaches 0 , the numerator is,

msinmx+nsinnx=msinm0+nsinn0=m0+

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