Evaluate :
`lim_( n -> oo ) (f(a + 1/n ) / f(a) )^(1/n ) = 1`

evaluate lim_ (n rarr oo) ((e ^ (n)) / (pi)) ^ ((1) / (n))

lim_ (n rarr oo) ((a-1 + b ^ ((1) / (n))) / (a)) ^ (n)

lim_ (n rarr oo) (2 ^ (n) -1) / (2 ^ (n) +1)

lim_ (n rarr oo) (2 ^ (n) -1) / (2 ^ (n) +1)

If f : R to R is given f (x) = x +1, then the value of lim _( b - oo) (1)/(n) [ f (0) + f ((5)/(n)) + f ((10)/(n)) +...+ f ((5 ( n -1))/( n )) ], is :

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) (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))

Let f be differentiable at x=a\ a n d\ l e t\ f(a)!=0. Evaluate (lim)_(n->oo){(f(a+1/n))/(f(a))}^ndot

Evaluate lim_ (n rarr oo) [(1+ (1) / (n)) 91+ (2) / (n)) .... (1+ (n) / (n))] ^ ((1) /(a))