Evaluate :

`lim_( n -> oo ) n/( n^2 + n )`

Evaluate: lim_ (n rarr oo) (1 ^ (4) + 2 ^ (4) + 3 ^ (4) + ... + n ^ (4)) / (n ^ (5)) - lim_ (n rarr oo) (1 ^ (3) + 2 ^ (3) + ... + n ^ (3)) / (n ^ (5))

Evaluate: lim_ (n rarr oo) (1 ^ (4) + 2 ^ (4) + 3 ^ (4) + ... + n ^ (4)) / (n ^ (5)) - lim_ (n rarr oo) (1 ^ (3) + 2 ^ (3) + ... + n ^ (3)) / (n ^ (5))

Evaluate: lim_ (n rarr oo) (1 * 2 + 2 * 3 + 3 * 4 + ... + n (n + 1)) / (n ^ (3))

Evaluate: lim_(n rarr oo)n^(-n^(2)){(n+2^(0))n+2^(-1))(n+2^(-2))(n+2^(-n+1))}^(n)

Evaluate: lim_(x rarr oo)sin^(n)((2 pi n)/(3n+1)),n in N

Evaluate: (lim)_(n rarr oo)[(n!)/(n^(n))]^(1/n)

Using Sandwich theorem, evaluate lim_ (n rarr oo) ((1) / (1 + n ^ (2)) + (1) / (2 + n ^ (2)) + ...... + (n) / (n + n ^ (2)))

Evaluate: lim_(n rarr oo)(n^(p)sin^(2)(n!))/(n+1)

Evaluate lim_ (n rarr oo) (1) / (n) sum_ (r = n + 1) ^ (2n) log_ (e) (1+ (r) / (n))