Trigonometric and exp limit 
Evaluation of $$\lim_{x\rightarrow \frac{\pi}{2}}\frac{\sin x-(\sin x)^{\sin x}}{1-\sin x+\ln (\sin x)}$$

Without Using L hopital Rule and series expansion. 
$\bf{My\; Try::}$ I have solved it using L hopital Rule and series expansion.
But I did not undersand How can i solve it Without Using L hopital Rule and series expansion
Help required, Thanks

It is not a direct answer.
By substition $x-\frac { \pi  }{ 2 } =t$ $$f\left( x \right) =\frac { \cos { x } -{ \cos { x }  }^{ \cos { x }  } }{ 1-\cos { x } +\ln { \cos { x }  }  } $$
$$\lim _{ t\rightarrow 0 }{ \frac { \sin { \left( \frac { \pi  }{ 2 } +t \right) - } { \sin { \left( \frac { \pi  }{ 2 } +t \right)  }  }^{ \sin { \left( \frac { \pi  }{ 2 } +t \right)  }  } }{ 1-\sin { \left( \frac { \pi  }{ 2 } +t \right) +\ln { \left( \sin { \left( \frac { \pi  }{ 2 } +t \right)  }  \right)  }  }  }  } =\lim _{ t\rightarrow 0 }{ \frac { \cos { t } -{ \cos { t }  }^{ \cos { t }  } }{ 1-\cos { t } +\ln { \cos { t }  }  }  } $$

Analysing graph of the function i think (i hope) we can get an answer
