function asymptotic where $f(x) = \frac{a + O(\frac{1}{\sqrt{x}})}{b + O(\frac{1}{\sqrt{x}})}$ If $a$ and $b$ are positive real numbers, and if $f(x)$ has the following asymptotic property
$f(x) = \frac{a + O(\frac{1}{\sqrt{x}})}{b + O(\frac{1}{\sqrt{x}})}$
then is the following true?
$f(x) = \frac{a}{b} + O(\frac{1}{\sqrt{x}})$
This might look like homework but it isn't.

Yes.  One way to see this is to actually do the long division (like the kind you learned in elementary school)!  Unfortunately, typesetting that in full on this forum will overtax my LaTeX powers.
Anyway, dividing $b + O\left(\frac{1}{\sqrt{x}}\right)$ into $a + O\left(\frac{1}{\sqrt{x}}\right)$ yields $\frac{a}{b}$ with a remainder of $O\left(\frac{1}{\sqrt{x}}\right)$.  So we have 
$$\frac{a + O\left(\frac{1}{\sqrt{x}}\right)}{b + O\left(\frac{1}{\sqrt{x}}\right)} = \frac{a}{b} + \frac{O\left(\frac{1}{\sqrt{x}}\right)}{b + O\left(\frac{1}{\sqrt{x}}\right)} = \frac{a}{b} + O\left(\frac{1}{\sqrt{x}}\right),$$
since $b + O\left(\frac{1}{\sqrt{x}}\right) = O(1).$
