Help me solve an exercise regarding limits

[heo83]

how to find limits of x to the power of sin x

I have this exercise:

         
      1 - [cos(x)]^sin(x)
lim --------------------- = ?????
x->0 tan(x)^3

(explain to someone that would be confused by these formulas: cos(x) to the power of sin(x), tan(x) to the power of 3)

Does anyone have the solution, please post it. Thanks a lot.
Besides can you explain it in details ?
Something like
>limit((1-cos(x)^sin(x))/tan(x)^3, x = 0); is not welcome. (that means I can use Maple, or computer to solve this exercise, to have the numeric value but I want to understand why and how?)
To me:
I tried to use l'Hospital but diff(cos(x)^sin(x),x) would cause a lot things relate to log that could make this exercise more difficult. Halt!

 

[zorro]

Re: Question on limits

Express cos(x) as exp(ln(cos(x)); then cos(x)^sin(x)=exp(sin(x)*ln(cos(x))).
Then apply L’Hopital rule twice for solve the indetermination.
The result is 1/2 .

Z