How did they cancel out O-terms in this fraction?

Question

While reading a book about algorithms, I came across this derivation:

$$ \frac{2a_0(2N) \ln(2N) + O(2N)}{2a_0N\ln N+O(N)} = \frac{2\ln(2N) + O(1)}{\ln N+O(1)} = 2 + O\left(\frac{1}{\log N}\right). $$

Could somebody please explain to me how we went from the left to the right?