Question

How do you evaluate the limit `(1-cosx)/tanx` as x approaches `0`?

Answer 1

`0`

Explanation:

Using de Moivre's identity

`e^(ix) = cos x+i sin x`

`(1-cosx)/tanx = (1-(e^(ix)+e^(-ix))/2)(e^(ix)+e^(-ix))/(e^(ix)-e^(-ix))=`
`((2-(e^(ix)+e^(-ix))))/(2(e^(ix)-e^(-ix)))(e^(ix)+e^(-ix))=`
`((e^(ix/2)-e^(-ix/2))^2(e^(ix)+e^(-ix)))/(2(e^(ix/2)+e^(-ix/2))(e^(ix/2)-e^(-ix/2))) =`
`=((e^(ix/2)-e^(-ix/2))(e^(ix)+e^(-ix)))/(2(e^(ix/2)+e^(-ix/2)))`

So

`lim_(x->0)(1-cosx)/tanx = lim_(x->0)((e^(ix/2)-e^(-ix/2))(e^(ix)+e^(-ix)))/(2(e^(ix/2)+e^(-ix/2)))=0`