Calculate the limit n^2/( 1 + 2 + 3 + ... + n ), x->infinity

To evaluate the limit of the rational function, when n tends to +inf.,we'll factorize both, numerator and denominator.

In this case, first, we'll write the sum of the denominator:

1+2+...+n = n*(n+1)/2

We'll substitute the denominator, by the result of the sum, we'll factorize  by the highest power of n, which in this case is n^2.

We'll have:

lim n^2/( 1 + 2 + 3 + ... + n )  = lim n^2/lim (1+2+3+ ... +n)

lim n^2/lim (1+2+3+ ... +n) = lim 2*n^2/lim n*(n+1)

We'll open the brackets from the denominator:

lim 2*n^2/lim n^2(1+1/n)

We'll divide by n^2:

lim 2/lim (1+1/n) = 2/(1+0)=2