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If L=("lim")(nvecoo)(2x3^2x2^3x3^4..... ...

If `L=("lim")_(nvecoo)(2x3^2x2^3x3^4..... x2^(n-1)x3^n)^(1/((n^(2+1)))` , then the value of `L^4` is

A

`-(1)/(4)`

B

`(1)/(2)`

C

1

D

2

Text Solution

Verified by Experts

The correct Answer is:
`(6)`
Clearly, n is even. Then,
`underset(ntooo)lim"("2^(1+3+5+...+n//2" terms ").3^(2+4+6+...+n//2" terms")")"^((1)/((n^(2)+1)))`
`=underset(ntooo)lim(2^((n^(2))/(4)).3^((n(n+2))/(4)))^((1)/((n^(2)+1)))`
`=2^(underset(ntooo)lim(1)/(4(1+(1)/(n^(2))))).3^(underset(ntooo)lim((1+(2)/(n)))/(4(1+(1)/(n^(2)))))`
`=2^((1)/(4))3^((1)/(4))=(6)^((1)/(4))`
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CENGAGE-LIMITS-Exercise (Numerical)
  1. If ("lim")(xvecoo)f(x) exists and is finite and nonzero and if ("lim"...

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  2. If L-("lim")(xvec2)((10-x)^(1/3)-2)/(x-2),t h e nt h ev a l u eof|1(4L...

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  3. If lim(xto0) (p sin2x+(1-cos2x))/(x+tanx)=1, then the value of p is.

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  4. The value of lim(xtooo) ((100)/(1-x^(100))-(50)/(1-x^(50))) is .

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  5. If L-("lim")(xvec2)((10-x)^(1/3)-2)/(x-2),t h e nt h ev a l u eof|1(4L...

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  6. The value of underset(xtooo)lim((100)/(1-x^(100))-(50)/(1-x^(50))) is ...

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  7. The value of ("lim")(xvecoo)[3sqrt((n+1)^2) -3sqrt((n-1)^2 ))] is

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  8. If L=("lim")(nvecoo)(2x3^2x2^3x3^4..... x2^(n-1)x3^n)^(1/((n^(2+1))) ,...

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  9. The value of ("lim")(xvecoo)((log)e((log)e x)/(e^(sqrt(x)))) is

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  10. about to only mathematics

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  11. The value of lim(x to oo ) (x-x^(2)log(e)(1+(1)/(x))) is .

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  12. Let S(n)=1+2+3+...+n " and " P(n)=(S(2))/(S(2)-1).(S(3))/(S(3)-1).(S(4...

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  13. If ("lim")(xvec1)(asin(x-1)+bcos(x-1)+4)/(x^2-1)=-2,t h e n|a+b| is

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  14. Let ("lim")(x to1)(x^a-a x+a-1)/((x-1)^2)=f(a)dot Then the value of f(...

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  15. Evaluate underset(xto1)lim((2)/(1-x^(2))-(1)/(1-x)).

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  16. If: ("lim")(xvec1)(1+a x+b x^2)^(c/((x-1)))=e^3, then the value ofb c ...

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  17. Let f ''(x) be continuous at x=0 If ("lim")(xvec0)(2f(x)-3a(f(2x)+bf(...

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  18. If L=("lim")(xvec0)(e^-(x^(2)/2)-cosx)/(x^3sinx),t h e nt h ev a l u ...

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  19. The integer n for which ("lim")(xvec0)((cosx-1)(cosx-ehatx)/(x^n) is f...

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  20. If ("lim")(xvec0)[1+x+(f(x))/x]^(1/x)=e^3, then find the value of 1n((...

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