Limits Doubt -

ITS

(xn^k/n)ⁿ n^k/n = 1

so its xⁿ = 0

if n is an infinte integer as

in case of - value of x , (-1)ⁿ is defined only for integers

ummm... how did u write the first line = 1 ?

n^k/n = e^klogn/n

now when n is infinity

logn/n is 0 ( use l'hospitals)

so its virtually e⁰ = 1

lim n→∞ xⁿ / n^-k

apply L'Hopital rule k times

limit = xⁿ(lnx)^k / (-1)^kk!

whose limiting value is 0 as n tends to infinity