$\underset{x\to 1}{\lim }\frac{(1-x)\left(1-x^2\right)\cdots \cdots \left(1-x^{2n}\right)}{{\left\{(1-x)\left(1-x^2\right)\cdots \cdots \left(1-x^n\right)\right\}}^2} \forall n\in N$

$\underset{x\to 1}{\lim }\frac{{\displaystyle \frac{(1-x)\left(1-x^2\right)\cdots \cdots \left(1-x^{2n}\right)}{{\left(1-x\right)}^{2n}}}}{{\displaystyle \frac{{\left\{(1-x)\left(1-x^2\right)\cdots \cdots \left(1-x^n\right)\right\}}^2}{{\left(1-x\right)}^{2n}}}}$

$\underset{x\to 1}{\lim }\frac{{\displaystyle \frac{1-x}{1-x}}\times \left({\displaystyle \frac{1-x^2}{1-x}}\right).............\times \left({\displaystyle \frac{1-x^{2n}}{1-x}}\right)}{{\left(\frac{1-x}{1-x}\times \frac{1-x^2}{1-x}\times .................\times \frac{1-x^n}{1-x}\right)}^2}$

Apply $\text{L'hospital}$ rule

$\underset{x\to 1}{\lim }\frac{{\displaystyle \frac{-1}{-1}}\times {\displaystyle \frac{-2x}{-1}}\times ...........\times {\displaystyle \frac{-2n\times x^{2n-1}}{-1}}}{{\left({\displaystyle \frac{-1}{-1}}\times {\displaystyle \frac{-2x}{-1}}\times ................{\displaystyle \frac{-n{x}^{n-1}}{-1}}\right)}^2}$

$\underset{x\to 1}{\lim }\frac{1\times 2\times 3\times ..........\times 2n}{{\left(1\times 2\times 3\times ............n\right)}^2}$

$=\frac{2n!}{{\left(n!\right)}^2}={}^{2n}C_n$