<s>[INST] What is the limit of the following function? $\lim_{n\rightarrow \infty }\left ( n-1-2\left (\frac{\Gamma(n/2)}{\Gamma((n-1)/2)} \right )^2 \right )$
 [/INST] Note that
$$
{{\Gamma (n/2)} \over {\Gamma (n/2 - 1/2)}} = \left( {n/2 - 1/2} \right)^{\,\overline {\,1/2\,} } 
$$
where $x^{\,\overline {\,a\,}} $ denotes the Rising Factorial (rising Pochammer).
It is an increasing function for $1<n$.
Because of that and considering the rules for summing the exponents of the Rising Factorial, we have:
$$
\left( {n/2 - 1/2} \right) = \left( {n/2 - 1/2} \right)^{\,\overline {\,1\,} }  = \left( {n/2 - 1/2} \right)^{\,\overline {\,\,1/2\,\,} } \left( {n/2} \right)^{\,\overline {\,\,1/2\,\,} }  > \left( {\left( {n/2 - 1/2} \right)^{\,\overline {\,\,1/2\,\,} } } \right)^{\,2} 
$$
and
$$
\left( {n/2 - 1} \right) = \left( {n/2 - 1} \right)^{\,\overline {\,1\,} }  = \left( {n/2 - 1} \right)^{\,\overline {\,\,1/2\,\,} } \left( {n/2 - 1/2} \right)^{\,\overline {\,\,1/2\,\,} }  < \left( {\left( {n/2 - 1/2} \right)^{\,\overline {\,\,1/2\,\,} } } \right)^{\,2} 
$$
which means
$$
\left( {n/2 - 1} \right) < \left( {\left( {n/2 - 1/2} \right)^{\,\overline {\,\,1/2\,\,} } } \right)^{\,2}  < \left( {n/2 - 1/2} \right)
$$
and therefore we know that the given function is bound between $0$ and $1$
$$
0 < n - 1 - 2\left( {\left( {n/2 - 1/2} \right)^{\,\overline {\,\,1/2\,\,} } } \right)^{\,2}  < 1\quad \left| {\;1 < n} \right.
$$
The Stirling series for the rising factorial is
$$
z^{\,\overline {\,w\,} }  \propto z^{\,w} \left( {1 + {{w\left( {w - 1} \right)} \over {2\,z}} + O\left( {{1 \over {z^{\,2} }}} \right)} \right)\quad \left| \matrix{
  \;\left| z \right| \to \infty  \hfill \cr 
  \;\left| {\arg (z + w)} \right| < \pi  \hfill \cr}  \right.
$$
so we get
$$
\eqalign{
  & \left( {n/2 - 1/2} \right)^{\,\overline {\,1/2\,} }  \propto \left( {n/2 - 1/2} \right)^{\,1/2\,} \left( {1 + {{\,1/2\,\left( {\,1/2\, - 1} \right)} \over {2\,\left( {n/2 - 1/2} \right)}} + O\left( {{1 \over {n^{\,2} }}} \right)} \right) =   \cr 
  &  = \left( {{{n - 1} \over 2}} \right)^{\,1/2\,} \left( {1 - {{\,1} \over {4\,\left( {n - 1} \right)}} + O\left( {{1 \over {n^{\,2} }}} \right)} \right) \cr} 
$$
and finally
$$
\eqalign{
  & n + 1 - 2\left( {\left( {n/2 - 1/2} \right)^{\,\overline {\,1/2\,} } } \right)^{\,2}  \propto n + 1 - \left( {n - 1} \right)\left( {1 - {{\,1} \over {2\,\left( {n - 1} \right)}} + O\left( {{1 \over {n^{\,2} }}} \right)} \right) =   \cr 
  &  = {1 \over 2} + O\left( {{1 \over n}} \right) \cr} 
$$
</s>