What is the limit of the following function? $\lim_{n\rightarrow \infty }\left ( n-1-2\left (\frac{\Gamma(n/2)}{\Gamma((n-1)/2)} \right )^2 \right )$
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 $1&lt;n$.
Because of that and considering the rules for summing the exponents of the Rising Factorial, we have:
$$
\left( {n/2 - 1/2} \right) = \left( {n/2 - 1/2} \right)^{\,\overline {\,1\,} }  = \left( {n/2 - 1/2} \right)^{\,\overline {\,\,1/2\,\,} } \left( {n/2} \right)^{\,\overline {\,\,1/2\,\,} }  &gt; \left( {\left( {n/2 - 1/2} \right)^{\,\overline {\,\,1/2\,\,} } } \right)^{\,2} 
$$
and
$$
\left( {n/2 - 1} \right) = \left( {n/2 - 1} \right)^{\,\overline {\,1\,} }  = \left( {n/2 - 1} \right)^{\,\overline {\,\,1/2\,\,} } \left( {n/2 - 1/2} \right)^{\,\overline {\,\,1/2\,\,} }  &lt; \left( {\left( {n/2 - 1/2} \right)^{\,\overline {\,\,1/2\,\,} } } \right)^{\,2} 
$$
which means
$$
\left( {n/2 - 1} \right) &lt; \left( {\left( {n/2 - 1/2} \right)^{\,\overline {\,\,1/2\,\,} } } \right)^{\,2}  &lt; \left( {n/2 - 1/2} \right)
$$
and therefore we know that the given function is bound between $0$ and $1$
$$
0 &lt; n - 1 - 2\left( {\left( {n/2 - 1/2} \right)^{\,\overline {\,\,1/2\,\,} } } \right)^{\,2}  &lt; 1\quad \left| {\;1 &lt; n} \right.
$$

The Stirling series for the rising factorial is
$$
z^{\,\overline {\,w\,} }  \propto z^{\,w} \left( {1 + {{w\left( {w - 1} \right)} \over {2\,z}} + O\left( {{1 \over {z^{\,2} }}} \right)} \right)\quad \left| \matrix{
  \;\left| z \right| \to \infty  \hfill \cr 
  \;\left| {\arg (z + w)} \right| &lt; \pi  \hfill \cr}  \right.
$$
so we get
$$
\eqalign{
  &amp; \left( {n/2 - 1/2} \right)^{\,\overline {\,1/2\,} }  \propto \left( {n/2 - 1/2} \right)^{\,1/2\,} \left( {1 + {{\,1/2\,\left( {\,1/2\, - 1} \right)} \over {2\,\left( {n/2 - 1/2} \right)}} + O\left( {{1 \over {n^{\,2} }}} \right)} \right) =   \cr 
  &amp;  = \left( {{{n - 1} \over 2}} \right)^{\,1/2\,} \left( {1 - {{\,1} \over {4\,\left( {n - 1} \right)}} + O\left( {{1 \over {n^{\,2} }}} \right)} \right) \cr} 
$$
and finally
$$
\eqalign{
  &amp; n + 1 - 2\left( {\left( {n/2 - 1/2} \right)^{\,\overline {\,1/2\,} } } \right)^{\,2}  \propto n + 1 - \left( {n - 1} \right)\left( {1 - {{\,1} \over {2\,\left( {n - 1} \right)}} + O\left( {{1 \over {n^{\,2} }}} \right)} \right) =   \cr 
  &amp;  = {1 \over 2} + O\left( {{1 \over n}} \right) \cr} 
$$
$$
\begin{aligned}
\lim_{n\rightarrow \infty }\left (n-1-2\left (\frac{\Gamma(n/2)}{\Gamma((n-1)/2)} \right )^2 \right)
&amp; = \lim _{n\rightarrow \:\infty \:}\:\left(n-1-2\left(\frac{\frac{2\left(\frac{n}{2}\right)!}{n}}{\frac{2\left(\frac{n-1}{2}\right)!}{n-1}}\right)^2\right)
\\&amp; = \lim _{n\rightarrow \:\infty \:}\:\left(n-1-2\left(\frac{\frac{2\left(\sqrt{2\pi \:\left(\frac{n}{2}\right)}\left(\frac{\left(\frac{n}{2}\right)}{e}\right)^{\frac{n}{2}}\right)}{n}}{\frac{2\left(\sqrt{2\pi \:\:\left(\frac{n-1}{2}\right)}\left(\frac{\left(\frac{n-1}{2}\right)}{e}\right)^{\frac{n-1}{2}}\right)}{n-1}}\right)^2\right)
\\&amp; = \lim _{n\rightarrow \:\infty \:}\:\left(n-1-\frac{n^{n-1}\left(n-1\right)^{-n+2}}{e}\right)
\\&amp; = \lim _{t\rightarrow 0\:}\:\left(\frac{1}{t}-1-\frac{\left(\frac{1}{t}\right)^{\left(\frac{1}{t}\right)-1}\left(\frac{1}{t}-1\right)^{-\frac{1}{t}+2}}{e}\right)
\\&amp; = \lim _{t\rightarrow 0\:}\:\left(\frac{e-\left(-t+1\right)^{\frac{2t-1}{t}}}{et}-1\right)
\\&amp; = \lim _{t\rightarrow 0\:}\:\left(\frac{e-\left(e-\frac{3et}{2}+o\left(t\right)\right)}{et}-1\right)
\\&amp; = \color{red}{\frac{1}{2}}
\end{aligned}
$$

Solved with Stirling's approximation and Taylor expansion