Find the limit $ \cos x / (1 - \sin x)^{2/3}$ My problem:

Find the following limit 
  $$\lim\limits_{x \to \pi/2} \frac{\cos x}{\left(1 - \sin x\right )^{2/3}}$$

I tried $(1 - \sin x)^2 = 1 - 2\sin x + \sin^{2}x$ but can go as far as that.

We have that by 


*

*$\cos x=\sin\left(\frac \pi 2 -x\right)$

*$\sin x=\cos\left(\frac \pi 2 -x\right)$
and $\frac \pi 2 -x =y \to 0$
$$\frac{\cos x}{\sqrt[3]{(1-\sin x)^2}} =\frac{\sin y}{\sqrt[3]{(1-\cos y)^2}}=\frac{\sin y}{y}\frac{\sqrt[3]{y^4}}{\sqrt[3]{(1-\cos y)^2}}\frac1{\sqrt[3]y} \to \pm \infty $$
indeed


*

*$\frac{\sin y}{y} \to 1$

*$\frac{\sqrt[3]{y^4}}{\sqrt[3]{(1-\cos y)^2}}=\sqrt[3]{\left(\frac{y^2}{{1-\cos y}}\right)^2}\to \sqrt[3] 2$

*$\frac1{\sqrt[3]y} \to \pm \infty$
