Find limit with factorial
We have:
$$ \begin{split} \lim_{n\to \infty} \frac{(2n-1)!}{(2n+1)!}=\lim_{n\to \infty} \frac{(2n-1)!}{(2n+1)*(2n)*(2n-1)!}&=\lim_{n \to \infty}\frac{1}{(2n+1)(2n)} \\ &=\lim_{n\to \infty}\frac{1}{4n^2+2n} \\ &=0\end{split} $$
Notice, expand the factorial as follows $$\lim_{n\to \infty}\frac{(2n-1)!}{(2n+1)!}$$ $$=\lim_{n\to \infty}\frac{(2n-1)!}{(2n+1)\cdot (2n)\cdot (2n-1)!}$$ $$=\lim_{n\to \infty}\frac{1}{(2n+1)\cdot (2n)}$$ $$=\lim_{n\to \infty}\frac{1}{2n+1}\cdot \lim_{n\to \infty}\frac{1}{2n}$$ $$=(0)(0)=\color{red}{0}$$