What is the limit of the following function? $\lim_{n\rightarrow \infty }\left ( n-1-2\left (\frac{\Gamma(n/2)}{\Gamma((n-1)/2)} \right )^2 \right )$

$$
\begin{aligned}
\lim_{n\rightarrow \infty }\left (n-1-2\left (\frac{\Gamma(n/2)}{\Gamma((n-1)/2)} \right )^2 \right)
& = \lim _{n\rightarrow \:\infty \:}\:\left(n-1-2\left(\frac{\frac{2\left(\frac{n}{2}\right)!}{n}}{\frac{2\left(\frac{n-1}{2}\right)!}{n-1}}\right)^2\right)
\\& = \lim _{n\rightarrow \:\infty \:}\:\left(n-1-2\left(\frac{\frac{2\left(\sqrt{2\pi \:\left(\frac{n}{2}\right)}\left(\frac{\left(\frac{n}{2}\right)}{e}\right)^{\frac{n}{2}}\right)}{n}}{\frac{2\left(\sqrt{2\pi \:\:\left(\frac{n-1}{2}\right)}\left(\frac{\left(\frac{n-1}{2}\right)}{e}\right)^{\frac{n-1}{2}}\right)}{n-1}}\right)^2\right)
\\& = \lim _{n\rightarrow \:\infty \:}\:\left(n-1-\frac{n^{n-1}\left(n-1\right)^{-n+2}}{e}\right)
\\& = \lim _{t\rightarrow 0\:}\:\left(\frac{1}{t}-1-\frac{\left(\frac{1}{t}\right)^{\left(\frac{1}{t}\right)-1}\left(\frac{1}{t}-1\right)^{-\frac{1}{t}+2}}{e}\right)
\\& = \lim _{t\rightarrow 0\:}\:\left(\frac{e-\left(-t+1\right)^{\frac{2t-1}{t}}}{et}-1\right)
\\& = \lim _{t\rightarrow 0\:}\:\left(\frac{e-\left(e-\frac{3et}{2}+o\left(t\right)\right)}{et}-1\right)
\\& = \color{red}{\frac{1}{2}}
\end{aligned}
$$
Solved with Stirling's approximation and Taylor expansion
