calculus and analysis – Limit if the limit is a function

Can MMA find limits if the limit can be expressed as a function?

Example:

$$lim_{x to infty}frac{Gammaleft(frac{x+1}{2}right)}{Gammaleft(frac{x}{2}right)} =sqrtfrac{x}{2}=infty$$
$\\$

Limit(Gamma((x + 1)/2) / Gamma(x/2), x -> (Infinity))

returns $infty$ but I am interested also in the more detailed answer $sqrtfrac{x}{2}$.

Only in case I presume the answer I can check that its true:

Limit(Gamma((x + 1)/2) / Gamma(x/2) - Sqrt(x/2), x -> (Infinity)) returns $0$.