Tough limit to evaluate I am trying to solve this limit problem
$$\lim_{x\to 1} {(1-x)(1-x^2)....(1-x^{2n})\over[(1-x)(1-x^2)....(1-x^n)]^2}$$
I am not able to figure how to to convert it to a compact form. Any tips?

The $r(1\le r\le n)$th term is $$T_r=\lim_{x\to1}\frac{(1-x^{2r-1})(1-x^{2r})}{(1-x^r)^2}$$
$$=\lim_{x\to1}\frac{x^{2r-1}-1}{x-1}\cdot\lim_{x\to1}\frac{x^{2r}-1}{x-1}\frac1{\left(\lim_{x\to1}\dfrac{x^r-1}{x-1}\right)^2}$$
Now for integer $\displaystyle a>-1,\lim_{x\to1}\dfrac{x^a-1}{x-1}=\dfrac{d(x^a)}{dx}_{(\text{at }x=1)}=\cdots=a$
or $\displaystyle \lim_{x\to1}\dfrac{x^a-1}{x-1}=\lim_{x\to1}(1+x+x^2+\cdots+x^{a-1})=a$
$$\implies T_r=\frac{(2r-1)2r}{r^2}$$
Hope you can take it home from here
