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

Answer

`1/e`

Explanation

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`