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

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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?
 
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1 - cos(x) = 2sin^2(x/2)

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

Using L'Hopitals on this does not lead to 0/0.
 
So I was missing something, thanks.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...

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