the limit has to be zero for it to be a null sequence. so isn't a null sequence
going back to what i said about the ratio of polynomials.
here are some rules you should get familiar with:
lets say you are given a ratio of polynomials and you want to find the infinite limit.
if the highest degree of the numerator is the same of the denominator, then the limit as to the ratio of the coefficients. (i hope you noted that when you got 1/4000)
the the degree in the denominator is greater than that of the numerator, then the ratio goes to zero as
if the degree of the numerator is greater than that of the denominator, then the limit as goes to , depending on the sign of the polynomials as x gets large
the degrees of the original function, of course.
i am talking about this:
the highest power in the numerator and denominator is 2 for both. so the infinite limits, as , goes to the ratio of the coefficients of the highest power. namely 1 in the numerator, and 4000 in the denominator. hence you get 1/4000 as your limit