Evaluate: `lim_(nrarr0) (((2n)!)/(n!n^(n)))^(1/n)`

Explanation

Let `y= underset(n rarroo)("lim") (((2n)!)/(n!n^(n)))^(1/n)rArr lny=underset(nrarroo)(lim)1/nln(((2n)!)/(n!n^(n)))`
`= underset(nrarr0)(lim)1/nln((2n(n-1)(2n-2)"......"(n+1))/(n^(n)))`
`= underset(nrarroo)(lim)underset(r=1)overset(n)sum(1)/(n)[ln(1+r//n)]=underset(0)overset(1)intln(1+x)dx = (xln(1+x))_(0)^(1)-underset(0)overset(1)int(x)/(1+x)dx`
`= (xln(1+x))_(0)^(1)-(x-ln(1+x))_(0)^(1)=ln2-(1-ln2)=ln4//erArr y = 4//e`