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lim(xto1) ((1-x)(1-x^(2))...(1-x^(2n)))/...

`lim_(xto1) ((1-x)(1-x^(2))...(1-x^(2n)))/({(1-x)(1-x^(2))...(1-x^(n))}^(2)),` `n in N,` equals

A

`" "^(2n)P_(n)`

B

`" "^(2n)C_(n)`

C

`2n!`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
`underset(xto1)lim((1-x)(1-x^(2))...(1-x^(2n)))/({(1-x)(1-x^(2))...(1-x^(n))}^(2))`
`=underset(xto1)lim(((1-x)/(1-x))((1-x^(2))/(1-x))...((1-x^(2n))/(1-x)))/((((1-x)/(1-x))((1-x^(2))/(1-x))...((1-x^(n))/(1-x)))^(2))`
`=(1xx2xx3...(2n))/((1xx2xx3...n)^(2))=((2n)!)/(n!n!)=^(2n)C_(n)`
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CENGAGE-LIMITS-Exercise (Single)
  1. The value of lim(xto2) (((x^(3)-4x)/(x^(3)-8))^(-1)-((x+sqrt(2x))/(x-2...

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  2. If lim(xto-2^(-)) (ae^(1//|x+2|)-1)/(2-e^(1//|x+2|))=lim(xto-2^(+)) si...

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  3. lim(xto1) ((1-x)(1-x^(2))...(1-x^(2n)))/({(1-x)(1-x^(2))...(1-x^(n))}^...

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  4. The value of lim(xto(1)/(sqrt(2))) (x-cos(sin^(-1)x))/(1-tan(sin^(-1)x...

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  5. Among (i) lim(xtooo) sec^(-1)((x)/(sinx))" and "(ii) lim(xtooo) sec^(-...

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  6. lim(xtooo) ((x^(3))/(3x^(2)-4)-(x^(2))/(3x+2))" is equal to "

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  7. lim(ntooo) (n(2n+1)^(2))/((n+2)(n^(2)+3n-1))" is equal to "

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  8. lim(xtooo) ((2x+1)^(40)(4x+1)^(5))/((2x+3)^(45)) is equal to

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  9. lim(xtooo) [sqrt(x+sqrt(x+sqrt(x)))-sqrt(x)] is equal to

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  10. lim(xtooo) (2+2x+sin2x)/((2x+sin2x)e^(sinx)) is equal to

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  11. lim(xtooo) ((x+1)^(10)+(x+2)^(10)+...+(x+100)^(10))/(x^(10)+10^(10)) i...

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  12. lim(xtooo) (2sqrt(x)+3root(3)(x)+4root(4)(x)+...+nroot(n)(x))/(sqrt((2...

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  13. If lim(ntooo) (n.3^(n))/(n(x-2)^(n)+n.3^(n+1)-3^(n))=1/3, then the ran...

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  14. lim(ntooo) n^(2)(x^(1//n)-x^(1//(n+1))),xgt0, is equal to

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  15. Let f(x)=lim(ntooo) (1)/(((3)/(pi)tan^(-1)2x)^(2n)+5). Then the set of...

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  16. f(x)=("ln"(x^(2)+e^(x)))/("ln"(x^(4)+e^(2x))). Then lim(x to oo) f(x) ...

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  17. The value of lim(ntooo) [(2n)/(2n^(2)-1)"cos"(n+1)/(2n+1)-(n)/(1-2n).(...

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  18. If f(x)=0 is a quadratic equation such that f(-pi)=f(pi)=0 and f((pi)/...

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  19. lim(xto1) (xsin(x-[x]))/(x-1), where [.] denotes the greatest integer ...

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  20. lim(xtooo) (x^(2)"tan"(1)/(x))/(sqrt(8x^(2)+7x+1)) is equal to

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