In my notes it was given that as $x\rightarrow 0,$
$\frac{x^{3/2}}{1-\cos x} = \mathcal{O}(x^{-1/2})$
It didnt give any explanation, so I was wondering what would the "method"/"intuition" be behind it, to be able to deduce/find out that $\mathcal{O}(x^{-1/2})?$
And how would I then find the order of $\frac{x^{3/2}}{1+\sin x}$ as $x\rightarrow 0.$