a tricky one...!!!

Question

Limit x tending to 0  , [cos(sinx) - cosx] / [ 1 - (e^x)² ]²  ?

Answer 1

Direct substitution yields (1-1)/(1-1)² = 0/0 which is an indeterminate form that can be handled by l"Hôpital's rule.

 

Derivative of numerator is -sin(sinx)(cosx) + sinx

 

Derivative of denominator = 2[1-e^2x](-2e^2x)   and now you take the ratio of the individual limits of those, as x→0 and see what happens.  If still indeterminate, you can re-apply  l"Hôpital's rule.  Just be careful because the chain rule has to be applied every time...