The value of limx- 11-x1-x2-1-x2n 1-x 1-x2-1-xn 2- n- is

Lim_x→ 1(1 x)(1 x^2)⋯(1 x^2n){ (1 x) (1 x^2)⋯(1 x^n) }^2 nbsp; = lim_x→ 1 ( 1 x1 x ) ( 1 x^21 x )⋯ ( 1 x^2n1 x ) [ ( 1 x1 x ) ( 1 x^21 x )⋯ ( 1 x^n1 x ) ]^2 ...

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Standard XII

Mathematics

Limit

The value of ...

Question

The value of $lim x \to 1 \frac{(1 - x) (1 - x^{2}) \dots (1 - x^{2 n})}{{(1 - x) (1 - x^{2}) \dots (1 - x^{n})}^{2}}, n \in N$ is

^{2 n} P_{n}

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^{2 n} C_{n}

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(2 n)!

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n!

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Solution

The correct option is B $^{2 n} C_{n}$
$lim x \to 1 \frac{(1 - x) (1 - x^{2}) \dots (1 - x^{2 n})}{{(1 - x) (1 - x^{2}) \dots (1 - x^{n})}^{2}}$
$= lim x \to 2 \frac{(\frac{1 - x}{1 - x}) (\frac{1 - x^{2}}{1 - x}) \dots (\frac{1 - x^{2 n}}{1 - x})}{{[(\frac{1 - x}{1 - x}) (\frac{1 - x^{2}}{1 - x}) \dots (\frac{1 - x^{n}}{1 - x})]}^{2}}$
$= \frac{1 \times 2 \times 3 \times \dots \times (2 n)}{(1 \times 2 \times 3 \times \dots \times n)^{2}}$
$= \frac{(2 n)!}{n! n!} =^{2 n} C_{n}$

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The value of limx→1(1−x)(1−x2)⋯(1−x2n){(1−x)(1−x2)⋯(1−xn)}2, n∈N is

The correct option is B 2nCnlimx→1(1−x)(1−x2)⋯(1−x2n){(1−x)(1−x2)⋯(1−xn)}2 =limx→2(1−x1−x)(1−x21−x)⋯(1−x2n1−x)[(1−x1−x)(1−x21−x)⋯(1−xn1−x)]2 =1×2×3×⋯×(2n)(1×2×3×⋯×n)2 =(2n)!n! n!=2nCn

The value of $lim x \to 1 \frac{(1 - x) (1 - x^{2}) \dots (1 - x^{2 n})}{{(1 - x) (1 - x^{2}) \dots (1 - x^{n})}^{2}}, n \in N$ is