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Evaluate the following limit: lim(nto ...

Evaluate the following limit:
`lim_(nto oo)[(n!)/(n^(n))]^(1//n)`

A

e

B

`1/e`

C

`pi/4`

D

`4/pi`

Text Solution

Verified by Experts

The correct Answer is:
B
Let `P=underset(nrarroo)(lim)[(n!)/(n^(n))]^(1//n)=underset(nrarroo)(lim)[(1)/(n).(2)/(n).(3)/(n).(4)/(n).(n)/(n)]^(1//n)`
`therefore" "logP=underset(nrarroo)(lim)(1)/(n)[log.(1)/(n)+log.(2)/(n)+…+log.(n)/(n)]`
`rArr" "logP=underset(nrarroo)(lim)sum_(=1)^(n)(1)/(n)log(r)/(n)`
`rArr" "logP=int_(0)^(1)log x dx=x [xlogx-x]_(0)^(1)=-1`
`rArr" "P=e^(-x)=(1)/(e)`
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MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS-DEFINITE INTEGRALS-PRACTICE EXERCISE (Exercise 2) (MISCELLANEOUS PROBLEMS)
  1. If int (0)^(pi//2) sin^(6) dx = (5pi)/32 , then the value of int (...

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  2. int (0)^(pi//4) [ sqrt(tan x)+ sqrt(cot x)] dx is equal to

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  3. Evaluate the following limit: lim(nto oo)[(n!)/(n^(n))]^(1//n)

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  4. Let a, b and c be non - zero real numbers such that int (0)^(3) (3ax...

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  5. The value of integral sum (k=1)^(n) int (0)^(1) f(k - 1+x) dx is

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  6. Ifint(sinx)^1t^2f(t)dt=1=1-s inx ,w h e r ex in (0,pi/2), then find t...

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  7. The value of int (2)^(4) { |x-2|+|x-3|} dx is

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  8. int (-1)^(2) |x|^(3) dx is equal to

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  9. int (0)^(3) (3x+1)/(x^(2)+9) dx =

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  10. int (0)^(pi//4) ( 4 sin 2 theta d theta )/(sin^(4) theta +cos^(4) thet...

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  11. The value of the integral int (-a)^(a) (xe^(x^(2)))/(1+x^(2)) dx is

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  12. int (-pi//2)^(pi//2) (dx)/(1+cosx) is equal to

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  13. The value of int (0)^(12a) (f(x))/(f(x)+f(12a-x))dx is

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  14. If kint(0)^(1)xf(3x)dx=int(0)^(3)tf(t)dt, then the value of k is

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  15. lim(nto oo)+(1)/(sqrt(n^(2)+n))+(1)/(sqrt(n^(2)+2n))+...(1)/(sqrt(n^(2...

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  16. lim(n->oo) (1^p+2^p+3^p+...........+n^p)/n^(p+1)

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  17. Evaluate: ("lim")(nvecoo)(1/(sqrt(4n^2-1))+1/(sqrt(4n^2-2^2))++1/(sqrt...

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  18. If f(x) = tanx-tan ^(3) x + tan^(5) x - tan ^(7) x + ... infty for o...

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  19. The value of int(0)^((pi)/(8))cos^(3)4 theta d theta is equal to -

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  20. int (0)^(pi//3) (cos x + sin x)/(sqrt(1+sin 2x))dx is equal to

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