Limit of x/sqrt(1-cosx) Approaching 0 from Negative Side

[aquitaine]

Ok, the problem is find the limit of x/sqrt(1-cosx) as x approaches 0 from the negative side.

First I tried simply applying l'hopital's rule to see what would happen, and it didn't work.

Next I tried rationalizing it by multiplying the numerator and denominator by sqrt(1+cosx), then using a trig identity (1-(cosx)^2=(sinx)^2 to get sqrt((sinx)^2) or simply sinx. Then I applied l'hopital's rule and ended up with a big mess that still ended up with 0/0.

Did I miss something or do something incorrectly?

 

[Vid]

1 - cos(x) = 2sin^2(x/2)

x/sqrt(1-cos(x)) = x/sqrt(2)sin(x/2)

Using l'hospital's on this does not lead to 0/0.

 

[aquitaine]

So I was missing something, thanks.