Evaluate :

F(x) = `lim_( n -> oo ) x^n /( 1 + x^n . e^x )`

lim_ (n rarr oo) (x ^ (n)) / (n!)

Evaluate: lim_(x rarr oo) [(4x^(n)+1)/(5x^(n)+5)] .

let f(x)=lim_(n rarr oo)(x^(2n)-1)/(x^(2n)+1)

If f (x) = lim _( n to oo) (n (x ^(1//n)-1)) for x gt 0, then which of the following is/are true?

lim_(n rarr oo)(1+(x)/(n))^(n)

Find the values of a if f(x)=(lim)_(n rarr oo)(ax^(2n)+2)/(x^(2n)+a+1) is continuous at x=1

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

Let f (x) = lim_ (n rarr oo) sum_ (r = 0) ^ (n-1) (x) / ((rx + 1) {(r + 1) x + 1})

Discuss the continuity of the function f(x) = lim_(n rarr oo) (log (2 + x)-x^(2n) sin x)/(1+x^(2n))"at x" = 1