Prove that `lim_(n->infty) (n^2+1)/(2n^2+3)=1/2`

Prove that lim_ (n rarr oo) ((1 ^ (2)) / (n ^ (3)) + (2 ^ (2)) / (n ^ (3)) + (3 ^ (2)) / ( n ^ (3)) + .... + (n ^ (2)) / (n ^ (3))) = (1) / (3)

The value of lim_(n rarr infty) (1)/(n) {(n+)(n+2)(n+3)…(n+n)}^(1//n) is equal to

Prove that lim_(n rarr oo)a_(n)=-(1)/(4), where a_(n)=n^(2),(sqrt(1+(1)/(n))+sqrt(1-(1)/(n))-2) for n in N.

Evaluate: lim_(n rarr infty)[1/n+1/(n+1)+1/(n+2)+ cdot cdot cdot +1/(3n)] .

The value of lim_(ntooo) [(2n)/(2n^(2)-1)"cos"(n+1)/(2n+1)-(n)/(1-2n).(n)/(n^(2)+1)] is

Let L= lim_(nrarr infty) int_(a)^(infty)(n dx)/(1+n^(2)x^(2)) , where a in R, then L can be

Consider the following statements : I. lim_(n rarr infty) (2^(n)+(-2)^(n))/(2^(n)) does not exist II. lim_(n rarr infty)(3^(n)+(-3)^(n))/(4^(n)) does not exist Then