lim _{x} {rightarrow 1} frac{(1-x)(1-x^2) cdo | vedclass

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(B) Let $L = \lim _{x \rightarrow 1} \frac{(1-x)(1-x^2) \cdots (1-x^{2n})}{\{(1-x)(1-x^2) \cdots (1-x^n)\}^2}$.We can rewrite the expression as:$L = \

Vedclass · Accepted Answer

$^{2n}C_n$

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(B) Let $L = \lim _{x ightarrow 1} \frac{(1-x)(1-x^2) \cdots (1-x^{2n})}{\{(1-x)(1-x^2) \cdots (1-x^n)\}^2}$.We can rewrite the expression as:$L = \lim _{x ightarrow 1} \frac{\prod_{k=1}^{2n} (1-x^k)}{\left(\prod_{k=1}^{n} (1-x^k)ight)^2}$.Using the property $\lim _{x ightarrow 1} \frac{1-x^k}{1-x} = k$,we divide the numerator and denominator by $(1-x)^{2n}$:$L = \lim _{x ightarrow 1} \frac{\prod_{k=1}^{2n} \frac{1-x^k}{1-x}}{\left(\prod_{k=1}^{n} \frac{1-x^k}{1-x}ight)^2}$.As $x ightarrow 1$,$\frac{1-x^k}{1-x} = 1 + x + x^2 + \dots + x^{k-1} ightarrow k$.Substituting these limits:$L = \frac{1 	imes 2 	imes 3 	imes \dots 	imes 2n}{(1 	imes 2 	imes 3 	imes \dots 	imes n)^2}$.$L = \frac{(2n)!}{(n!)^2} = \frac{(2n)!}{n! n!} = {}^{2n}C_n$.

$\lim _{x}$ ${\rightarrow 1} \frac{(1-x)(1-x^2) \cdots (1-x^{2n})}{\{(1-x)(1-x^2) \cdots (1-x^n)\}^2} = \dots, \forall n \in N$

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