`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.

Expert Answers
Neethu eNotes educator| Certified Educator

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`