`lim_(x->0) (cos(mx) - cos(nx))/(x^2)` 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...

`lim_(x->0) (cos(mx) - cos(nx))/(x^2)` 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.

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Chapter 4, 4.4 - Problem 32 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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nees101 | Student, Graduate | (Level 2) Adjunct Educator

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Given the limit `lim_{x->0}(cos(mx)-cos(nx))/x^2` . We have to find the limit value

Applying the limits we get,

`lim_{x->0}(cos(mx)-cos(nx))/x^2=0/0`

Using L'Hospital's rule and then applying the limit we get,``

`lim_{x->0}(-msin(mx)+nsin(nx))/(2x)=0/0`

So again using L'Hospital's rule we get,

`lim_{x->0}(-m^2cos(mx)+n^2cos(nx))/2=(n^2-m^2)/2`

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