I need help with a limit without using L'Hopital I need to do the following limit without using L'Hopital and I have not been able, please help

$$\lim\limits_{x \to 3} \left(\frac{x-1}{2x-4}\right)^{\frac{1}{x-3}}$$


Since $\lim _{ x\rightarrow 0 }{ { \left( 1+x \right)  }^{ \frac { 1 }{ x }  } } =e$ (or $\\ \lim _{ x\rightarrow \infty  }{ { \left( 1+\frac { 1 }{ x }  \right)  }^{ x } } =e\\ $)

$$\lim _{ x\to 3 } \left( \frac { x-1 }{ 2x-4 }  \right) ^{ \frac { 1 }{ x-3 }  }=\lim _{ x\to 3 } \left( 1+\frac { 3-x }{ 2x-4 }  \right) ^{ \frac { 1 }{ x-3 }  }=\\ =\lim _{ x\to 3 }{ \left[ \left( 1+\frac { 1 }{ \frac { 2x-4 }{ 3-x }  }  \right) ^{ \frac { 2x-4 }{ 3-x }  } \right]  } ^{ \frac { 3-x }{ 2x-4 } \frac { 1 }{ x-3 }  }=\lim _{ x\to 3 }{ { e }^{ -\frac { 1 }{ 2x-4 }  } } =\frac { 1 }{ \sqrt { e }  } $$ 


You can solve it with other way let say $x-3=z,$ then $$\lim _{ z\rightarrow 0 }{ \left( 1+\frac { -z }{ 2+2z }  \right) ^{ \frac { 1 }{ z }  } } =\lim _{ z\rightarrow 0 }{ { \left( \left( 1+\frac { 1 }{ -\frac { 2+2z }{ z }  }  \right) ^{ -\frac { 2+2z }{ z }  } \right)  }^{ -\frac { z }{ 2+2z } \frac { 1 }{ z }  } } =\lim _{ z\rightarrow 0 }{ { e }^{ -\frac { 1 }{ 2z+2 }  } } =\frac { 1 }{ \sqrt { e }  } $$
