Home
Class 12
MATHS
Evaluate the following limit: lim(nto ...

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

Text Solution

Verified by Experts

The correct Answer is:
`1/e`
Let `A=lim_(n to oo) [(n!)/(n^(n))]^(1//n)`
`=lim_(nto oo) ((1/xx2xx3……..n)/(nxnxxn……..n))^(1//n)`
`=lim_(nto oo)(1/nxx2/nxx3/n……..n/n)^(1//n)`
`:. logA=lim_(nto oo) 1/n["log" 1/n+"log"2/n+………+"log"n/n]`
`=lim_(nto oo) sum_(r=1)^(n)1/n"log"r/n`
`int_(0)^(1)logxdx=[x logx]_(0)^(1)-int_(0)^(1)x xx1/x dx`
`=0-int_(0)^(1)dx=0-[x]_(0)^(1)=-1`
`:.A=e^(-1)=1//e`
Promotional Banner

Topper's Solved these Questions

  • DEFINITE INTEGRATION

    CENGAGE ENGLISH|Exercise SCQ_TYPE|113 Videos

  • DEFINITE INTEGRATION

    CENGAGE ENGLISH|Exercise MCQ_TYPE|27 Videos

  • DEFINITE INTEGRATION

    CENGAGE ENGLISH|Exercise SOLVED EXAMPLE_TYPE|20 Videos

  • CURVE TRACING

    CENGAGE ENGLISH|Exercise EXERCISES|24 Videos

  • DETERMINANT

    CENGAGE ENGLISH|Exercise Multiple Correct Answer|5 Videos

Similar Questions

Explore conceptually related problems

Evaluate the following limits : Lim_( n to oo) (1+2/n)^(2n)

Evaluate the following limits : Lim_(n to oo) (Sigman^(3))/(n^(4))

Evaluate the following limit: (lim)_(n->oo)2^(n-1)sin(a/(2^n))

Evaluate the following limits : Lim_(n to oo) (1+2+3+...+n)/(n^(2)) ( or Lim_(x to oo) (Sigman)/(n^(2)))

Evaluate the following limit: lim_(nto oo)(sum_(r=1)^(n) sqrt(r)sum_(r=1)^(h)1/(sqrt(r)))/(sum_(r=1)^(n)r)

lim_(n to oo)(n!)/((n+1)!-n!)

Evaluate the following (i) lim_(n to oo)((1)/(n^(2))+(2)/(n^(2))+(3)/(n^(2))....+(n-1)/(n^(2))) (ii) lim_(n to oo)((1)/(n+1)+(1)/(n+2)+....+(1)/(2n)) (iii) lim_(n to oo)((n)/(n^(2)+1^(2))+(n)/(n^(2)+2^(2))+....+(n)/(2n^(2))) (iv) lim_(n to oo)((1^(p)+2^(p)+.....+n^(p)))/(n^(p+1)),pgt0

Evaluate the following limit: (lim)_(n->oo)((n+2)!+(n+1)!)/((n+2)!-(n+1)!)

Evaluate the following limit: (lim)_(n->oo)(1/(n^2)+2/(n^2)+3/(n^2)++(n-1)/(n^2\ ))

Evaluate the following limit: (lim)_(n->oo)(sin(a/(2^n)))/(sin(b/(2^(n\ ))))

CENGAGE ENGLISH-DEFINITE INTEGRATION -CAE_TYPE
  1. Evaluate ("lim")(nvecoo)sum(k=1)^nk/(n^2+k^2)

    Text Solution

    |

  2. Evaluate the following limit: lim(nto oo)(sum(r=1)^(n) sqrt(r)sum(r=...

    Text Solution

    |

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

    Text Solution

    |

  4. P rov et h a t4lt=int1^3sqrt(3+x^2)lt=4sqrt(3)

    Text Solution

    |

  5. If I1=int0^1 2^x^2,I2=int0^1 2^x^3dx ,I3=int1^2^x^2dx ,I4=int1^2 2^x^3...

    Text Solution

    |

  6. IfII=int0^(pi//2)cos(sinx)dx ,I2=int0^(pi/2)sin(cosx)d ,a n dI3=int0^(...

    Text Solution

    |

  7. Prove that pi/6<int0^1(dx)/(sqrt(4-x^2-x^3))<pi/(4sqrt(2))

    Text Solution

    |

  8. Evaluate int(0)^(pi//2)|sinx-cosx|dx.

    Text Solution

    |

  9. Evaluate: int(-1)^4f(x)dx=4a n dint2^4(3-f(x))dx=7, then find the val...

    Text Solution

    |

  10. Evaluate int(1)^(5)sqrt(x-2)sqrt(x-1)dx.

    Text Solution

    |

  11. Evaluate: int(-1)^3(tan^(-1)(x/(x^2+1))+tan^(-1)((x^2+1)/x))dx

    Text Solution

    |

  12. Evaluate int(1)^(a)x.a^(-[log(e)x])dx,(agt1).Here [.] represents the g...

    Text Solution

    |

  13. Evaluate int(1)^(e^(6))[(logx)/3]dx, where [.] denotes the greatest in...

    Text Solution

    |

  14. Find the value of int(-1)^1[x^2+{x}]dx ,w h e r e[dot]a n d{dot} denot...

    Text Solution

    |

  15. Evaluate:- int0^(pi)[cot x]dx ,w h e r e[dot] denotes the greatest in...

    Text Solution

    |

  16. Prove that int0^x[t]dt=([x]([x]-1))/2+[x](x-[x]), where [.] denotes ...

    Text Solution

    |

  17. Evaluate: int0^oo[2e^(-x)]dx ,w h e r e[x] represents greatest intege...

    Text Solution

    |

  18. If f(a+b-x)=f(x), then prove that inta^b xf(x)dx=(a+b)/2inta^bf(x)...

    Text Solution

    |

  19. The value of the integral int3^6 sqrtx/(sqrt(9-x)+sqrtx)dx is

    Text Solution

    |

  20. Find the value of int(0)^(1)root(3)(2x^(3)-3x^(2)-x+1)dx.

    Text Solution

    |